Transference for loose Hamilton cycles in random 3-uniform hypergraphs

A loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac’s theorem in mind, it is natural to ask what minimum d -degree condition guarantees the existence of a loose Hamilton cycle in a k -uniform hypergraph. For k “ 3 and each d P t 1 , 2 u , the necessary and sufficient such condition is known precisely. We show that these results adhere to a ‘transference principle’ to their sparse random analogues. The proof combines several ideas from the graph setting and relies on the absorbing method. In particular, we employ a novel approach of Kwan and Ferber for finding absorbers in subgraphs of sparse hypergraphs via a contraction procedure. In the case of d “ 2, our findings are asymptotically optimal.


Introduction
The question of deciding when a given graph is Hamiltonian is in general notoriously difficult and was included in Karp's original list of 21 NP-complete problems [34].Being a fundamental problem in graph theory (and computer science), Hamiltonicity has inspired a long line of work exploring sufficient conditions for it.Perhaps the best known among those is the classical theorem of Dirac [14]: every graph on n ě 3 vertices with minimum degree at least n{2 contains a Hamilton cycle.Another regime in which the problem is understood better than in the general case is that of random graphs (also, more broadly, quasi-random graphs and expanders).In that regard, Pósa [59] and independently Korshunov [43] proved that if p ě C log n{n, for some constant C ą 0, then the Erdős-Rényi binomial random graph Gpn, pq 1 is with high probability 2  Hamiltonian.The more precise value p " ppnq for which the former holds was later determined by Komlós and Szemerédi [42], and an even stronger, so-called hitting-time result, was shown by Ajtai, Komlós, and Szemerédi [1] and independently by Bollobás [7].
Inquiring further into properties of random graphs, Sudakov and Vu [63] asked how resilient Gpn, pq is with respect to having a Hamilton cycle.A graph G is α-resilient with respect to a property P if after the removal of at most an α-fraction of edges incident to every vertex of G, the resulting graph (still) contains P. Observe that Dirac's theorem states exactly that: the complete graph on n vertices K n is 1{2-resilient with respect to Hamiltonicity (with 1{2 being optimal as witnessed by two disjoint cliques of size n{2).The work of Sudakov and Vu initiated a systematic study of minimum degree requirements in the flavour of Dirac's theorem for random graphs as they showed that for p " log4 n{n, w.h.p.Gpn, pq is p1{2 ´op1qq-resilient with respect to Hamiltonicity.A full analogue of Dirac's theorem for random graphs was later established by Lee and Sudakov [51]: if p " log n{n then w.h.p.Gpn, pq is p1{2 ´op1qq-resilient with respect to Hamiltonicity.Even more, the absolutely best-possible hitting-time results were recently obtained by Montgomery [52] and independently Nenadov, Steger, and the second author [56].
There are, however, certain deficiencies of the basic notion of resilience in the usual binomial random graph-for example, it is unable to capture the behaviour of Gpn, pq with respect to containment of large structures riddled with triangles.Namely, as soon as p " op1q, one can easily prevent as many as Θpp ´2q vertices from being in a triangle while removing only opnpq edges incident to each vertex (see [3,31]).This makes the study of resilience for, e.g., a triangle-factor or any power of a Hamilton cycle futile.Recently, Fischer, Steger, Škorić, and the second author extended the notion of resilience to H-resilience in order to study robustness of Gpn, pq with respect to the containment of the square of a Hamilton cycle [21].Roughly speaking, they determined the smallest α P r0, 1s such that, for p " log3 n{ ?n, w.h.p. every G Ď Gpn, pq in which every vertex belongs to at least pα `op1qqp 3  `n 2 ˘triangles contains the square of a Hamilton cycle.
Such a concept naturally corresponds to resilience in hypergraphs, with the added advantage that edges in a random hypergraph are independent, unlike copies of H in Gpn, pq.Thus, the hypergraph counterpart to the questions above can be a solid test case for developing new ideas and techniques.Of course, studying resilience of Hamiltonicity in hypergraphs is also of independent interest as a natural 'high-dimensional' generalisation of one of the most important questions in graph theory to a radically more challenging setting.
In this paper, we are concerned with Dirac-type conditions, hence resilience, for Hamiltonicity in random k-uniform 3 hypergraphs.The notions of cycles and degrees do not generalise unambiguously to the hypergraph setting, so in what follows we make them more specific.For some 1 ď ℓ ă k, an ℓ-cycle in a k-graph is a cyclic sequence of vertices such that every vertex belongs to some edge, every edge consists of k consecutive vertices, and each two consecutive edges overlap in exactly ℓ vertices.The most studied special cases are ℓ " 1 and ℓ " k ´1, which are referred to as a loose and a tight cycle, respectively.Note that the number of vertices in any ℓ-cycle is necessarily divisible 4 by k ´ℓ.In a k-graph H, for some 1 ď d ă k, the d-degree deg H pSq of a d-element set of vertices S Ď V pHq is the number of edges A P EpHq such that S Ď A. The 1-degree deg H pvq of a vertex v is referred to simply as its degree, and the pk ´1q-degree of a pk ´1q-set as its codegree.The minimum d-degree δ d pHq of H is the minimum value among the d-degrees over all d-sets S Ď V pHq.
When it comes to (sparse) random structures, Dudek, Frieze, Loh, and Speiss [15] showed that H k pn, pq5 w.h.p. contains a loose Hamilton cycle, provided that p " log n{n k´1 (this generalises a previous result of Frieze [22]; see also [17] for a short proof and [58] for a much more general framework from which this follows directly).This result is asymptotically optimal, since for p of lower order of magnitude, H k pn, pq w.h.p. has isolated vertices.
Our main contribution is a resilience variant of this result for 3-graphs, thus transferring the formerly mentioned Dirac-type statements to the sparse random setting.
Theorem 1.1.Let d P t1, 2u.For every γ ą 0 there is a C ą 0 such that the following holds.Suppose p ě C log n ¨maxtn ´3{2 , n ´3`d u and n is even.Then H 3 pn, pq w.h.p. has the property that every spanning subgraph G Ď H 3 pn, pq with δ d pGq ě pδ d `γqp `n´d 3´d ˘contains a loose Hamilton cycle.
The constants δ 1 " 7{16 and δ 2 " 1{4 are optimal-in other words, w.h.p. there exists a subgraph G of H 3 pn, pq with minimum d-degree pδ d ´op1qq `n´d 3´d ˘which does not have a loose Hamilton cycle.One can see this by, e.g., considering analogues of dense extremal constructions from [10] (in case d " 1) and [47] (in case d " 2).
The more interesting part are the limitations for the density p. Observe that one can more concisely summarise the density requirements as p ě C log n{n 2´d{2 , however we chose to present them separately to underline their distinct origins.In case d " 2, if p " c log n{n for some small c ą 0, then w.h.p.H 3 pn, pq contains pairs of vertices with 2-degree equal to zero, and thus Theorem 1.1 is optimal (up to the multiplicative constant).However, the bound p ě Cn ´3{2 log n for d " 1 is 'artificial' but crucial for our proof technique.It is highly likely, but probably quite challenging to prove, that the result should remain true all the way down to p " Ωplog n{n 2 q-the threshold for appearance of a loose Hamilton cycle in H 3 pn, pq.It seems that one would need to explore completely different techniques in order to establish this.
Note that this provides a resilience result for H 3 pn, pq with respect to containment of a loose Hamilton cycle-it shows that H 3 pn, pq is p9{16 ´op1qq-resilient for this property (similarly one can think of 'codegree resilience' in terms of δ 2 ).This partially answers a question raised by Frieze [23,Problem 56] in his survey on Hamilton cycles in random graphs.Being only the beginning of the story, a natural next step would be an extension to hypergraphs of larger uniformities.One obstacle on the way is that the corresponding dense Dirac condition with d " 1 is not known for k-graphs with k ě 4 (a 'transference' result could be obtained even without knowing this value, see [20]).In addition, our methods do not seem to straightforwardly generalise to k ě 4 regardless of the type of degree considered.
Usually, establishing properties of hypergraphs is substantially more difficult than in the case of their graph counterparts, hence it is not surprising that not much is known about resilience of random hypergraphs.Prior to this, Clemens, Ehrenmüller, and Person [11] studied resilience of H k pn, pq with respect to the containment of a Hamilton Berge cycle, and, very recently, Allen, Parczyk, and Pfenninger [2] proved that H k pn, pq is resilient for containment of tight Hamilton cycles.On a related note, Ferber and Kwan [20] proved a very general 'transference principle' concerning perfect matchings in random k-graphs (see also [18] for a specific result when d " k ´1).
On the whole, our proof follows a standard strategy for embedding relying on the absorbing method.In a nutshell, this method allows one to reduce the problem of finding a spanning structure to the usually significantly easier one of finding an almost-spanning one.For the latter we combine several ideas originating from their graph counterparts such as the DFS technique for finding long paths (Section 4), path connection techniques (Section 5), and the sparse regularity method (Section 3).As always, the most difficult, involved, and creative part comes in designing and finding the absorber (see Definition 6.1 below).The 'finding' part is partially done through a contraction procedure of Ferber and Kwan which helps with finding absorbers inside a regular partition of H 3 pn, pq.We discuss this, as well as the absorbing method in general, in much greater detail in Section 6.All these ingredients are mixed together following a usual recipe to give a proof of Theorem 1.1 (Section 7).

Preliminaries
Our graph theoretic notation mostly follows standard textbooks in the area, e.g.[8].More specifically, for a (hyper)graph G " pV, Eq we let vpGq and epGq denote the number of its vertices and edges, respectively.Given a set of vertices X Ď V pGq, G ´X stands for the induced graph GrV pGq ∖ Xs.For sets X, Y, Z Ď V pGq we let e G pX, Y, Zq denote the number of triples px, y, zq P X ˆY ˆZ for which xyz P EpGq.Instead of e G ptxu, Y, Zq, we write e G px, Y, Zq for brevity.Similarly, if P Ď `V pGq 2 ˘is a set of pairs of vertices, then e G pP, Zq counts the number of (ordered) pairs ptx, yu, zq for which xyz P EpGq with tx, yu P P and z P Z.For a set W Ď V pGq, we use deg G px, W q, respectively deg G ptx, yu, W q, for the number of distinct pairs ty, zu with y, z P W , respectively the number of vertices z P W , for which xyz P EpGq.The neighbourhood N G pv, W q of a vertex v in a set W Ď V pGq refers to the set of vertices w P W such that uvw P G for some u P W , and we denote N G pv, V pGqq as N G pvq for brevity.Similarly, the neighbourhood N G ptv, uu, W q of a pair of vertices v, u in a set W Ď V pGq denotes the set of vertices w P W with uvw P G, and we abbreviate N G ptv, uu, V pGqq to N G pv, uq.
A loose path of length ℓ P N is an ordered sequence of distinct vertices v 1 , . . ., v 2ℓ`1 and ℓ edges: v 2i´1 v 2i v 2i`1 P EpGq for all i P rℓs.Note that a loose path always consists of an odd number of vertices.Throughout, whenever we say path we mean loose path and we write xy-path for a path whose start-and end-points are x and y.A (loose) cycle is defined similarly.
All logarithms are in base e.For n P N, rns stands for the set of first n integers, that is rns :" t1, . . ., nu.We use standard asymptotic notation o, O, ω, Ω, and Θ.For a set W and an integer d, `W d ˘denotes the collection of all d-element subsets, d-sets for brevity, of W , and W d denotes the collection of all distinct d-tuples pw 1 , . . ., w d q with w i P W and w i ‰ w j for each 1 ď i ă j ď d.When using set-theoretic notation, we treat tuples as corresponding sets, e.g.x P pw 1 , . . ., w d q stands for x P tw 1 , . . ., w d u and two tuples are disjoint if they do not share an element.

Distribution of edges
In this subsection we list some lemmas that give upper bounds on the number of edges between various vertex sets in H 3 pn, pq.Lemma 2.1.For every γ ą 0 and C ą 0 there exists λ ą 0 such that w.h.p.G " H 3 pn, pq satisfies the following, provided that n 2 p " log n.There are no two disjoint sets A, B Ď V pGq of sizes a and b such that 1 ď a ď λn, b ď Ca, and e G pA, B, V pGqq ě γap `n´1 This is at most e ´ωpa log nq provided λ is chosen small enough with respect to γ and C. Then the union bound over at most e pa`bq log n ď e 2Ca log n choices for the sets A and B completes the proof.

Expansion
The following lemma captures the fact that expansion of vertices behaves as expected in a not too sparse subgraph G of the random graph H 3 pn, pq.Namely, if a vertex v has the property that for some W the minimum degree of Grtvu Y W s is at least Ωpn 2 pq, then there are at least ?n vertices w 1 P W and Θpnq vertices w 2 P W for which there is a vw i -path of length i in Grtvu Y W s. It plays an important role both for proving the Connecting Lemma and finding absorbers.
Lemma 2.5.For every γ ą 0 there exist ξ, C ą 0 such that w.h.p.H 3 pn, pq satisfies the following, provided that p ě Cn ´3{2 log n.Let G Ď H 3 pn, pq and W Ď V pGq with δ 1 pGrW sq ě γp `n´1 2 ˘.For every x P V pGq for which deg G px, W q ě γp `n´1 x closes an edge with).Then, with X 1 denoting the vertices in P 1 x , we have On the other hand, by the property of Lemma 2.2 (ii ), assuming |X ∖ X 1 | ě ?n ě ε ´3 log n{pnpq and as |Q| " 6ξn (taking a superset if necessary), we have leading to a contradiction for ξ small enough.We just need to show that there is tu, vu P P 1 x and w 3 , w 4 P W ∖ Q which comprise an edge in G with v. Indeed, exactly the same computation as above establishes this, which completes the proof.

Degree inheritance properties
We first state a slightly strengthened version of [20,Lemma 5.5].The strengthening comes in the bound on p, which is stated to match the one from Theorem 1.1 (and in fact, most of the statements in this paper).For the case d " 2 we require p " Θplog n{nq, in contrast to Ωpn ´2 log 3 nq, and here the desired property is in fact easily obtained through Chernoff's inequality and the union bound due to independence.In case d " 1 the former requirement of p ě n ´2 log 3 n is actually necessary for this particular proof.
Lemma 2.6.For every γ, µ, ξ ą 0, there is a C such that w.h.p.H 3 pn, pq satisfies the following, provided that p ě C log n ¨maxtn ´3{2 , n ´3`d u.Let G Ď H 3 pn, pq and let W Ď V pGq be a uniformly random set of size at least ξn.Then w.h.p. every d-set S Ď V pGq that satisfies Proof.For d " 2, the codegree of a pair of vertices into W follows a hypergeometric distribution with mean Ωpnpq, for which Chernoff's inequality applies (see, e.g.[32,Theorem 2.10]).The statement thus follows directly from it and the union bound over all pairs of vertices.In case d " 1 the assertion holds by [20, Lemma 5.5], since p ě Cn ´3{2 log n ě n ´2 log 3 n.
The next lemma allows us to start with a graph in which almost all d-sets have at least some degree and pick a random subgraph of it such that in it, with positive probability, all d-sets have at least some (slightly smaller) degree.

Lemma 2.7 ([20, Lemma 3.4]).
There is a c ą 0 such that the following holds.Let d P t1, 2u and γ, δ, µ ą 0. Let G be an n-vertex 3-graph in which all but δ `n d ˘of the d-sets have degree at least pµ `γq `n´d 3´d ˘.Let S be a uniformly random subset of s ě 2d vertices of G. Then with probability at least 1 ´`s d ˘pδ `e´cγ 2 s q, the random induced subgraph GrSs has minimum d-degree at least pµ `γ{2q `s´d 3´d ˘.
The last two lemmas establish that in a subgraph of the random hypergraph H 3 pn, pq with sufficiently large minimum degree, after an adversary removes λn vertices, for some tiny λ ą 0, almost all d-sets still keep a significant portion of their original degree in the resulting graph.
Lemma 2.8.For every γ, µ ą 0 there exist λ, C ą 0 such that w.h.p.H 3 pn, pq satisfies the following, provided that p ě Cn ´3{2 log n.Let G Ď H 3 pn, pq with δ 1 pGq ě pµ `γqp `n´1 2 ˘and let S Ď V pGq be of size |S| ď λn.Then there exists a set T of size at most ?n such that for U :" V pGq∖pSYT q, the graph GrU s is of minimum degree at least pµ`γ{4qp `|U|´1 2 ˘.Furthermore, if for some x, y then T can be chosen to avoid x, y.
Proof.Initially, let T :" ∅.As long as there exists v P V pGq ∖ pS Y T q and v ‰ x, y with add such a vertex to T .Stop this process at the first point in time when |T | " ε ´3 log n{pnpq, for some small enough ε ą 0. Note that by the assumption on p, it also holds that |T | ă ?n.We then have On the other hand, as H 3 pn, pq w.h.p. satisfies the conclusion of Lemma 2.2 (ii ), which is a contradiction with the former, for λ ą 0 small enough.
Consider now x and its degree into V pGq ∖ pS Y T q.If it does not satisfy the bound promised by the lemma, this means On the other hand, by the property of Lemma 2.2 (i ), This is a contradiction with the former for C ą 0 large enough as n 3 p 2 ě C 2 log 2 n by the assumption on p.
This leads to a contradiction for λ ą 0 sufficiently small and C large enough, as |S| ď λn and np " log n.

(Hyper)graph theory
The following Dirac-type conditions for the existence of a loose Hamilton cycle were mentioned in the introduction but we state them here explicitly in the form in which we use them later.Recall, δ 1 " 7{16 and δ 2 " 1{4.The next lemma gives a minimum degree condition for a 3-graph that ensures that any pair of vertices is contained in some loose cycle of length three.We make use of it later for finding absorbers.
Lemma 2.11.Let H be a sufficiently large n-vertex 3-uniform hypergraph which satisfies δ 1 pHq ě 7  16 `n 2 ˘.Then for every pair of vertices u, v P V pHq, there exist distinct vertices a 1 , a 2 , a 3 , a 4 P V pHq ∖ tu, vu such that a 1 a 2 a 3 , a 2 a 4 u, a 3 a 4 v P EpHq, that is a 3 a 1 a 2 ua 4 va 3 is a loose cycle.
Since 7{16 " p5{8 `γq 2 for γ " ?7{4 ´5{8 ą 0, by the discussion above there is a W Ď V pHq such that, without loss of generality, deg H pu, wq ě γn and deg H pv, wq ě 3n{8 for all w P W . Pick an arbitrary a 4 P W and an arbitrary a 2 P V pHq ∖ tvu such that ua 4 a 2 P EpHq.Note that deg H pv, a 4 q ě 3n{8 and |N H pa 2 q| ě ?7n{4 due to the minimum degree condition for a 2 .Therefore, since ?7{4 `3{8 ą 1, we have |N H pv, a 4 q X N H pa 2 q| ą 2 (with plenty of room to spare).Pick a 3 P N H pv, a 4 q X N H pa 2 q ∖ tuu and pick a 1 P N H pa 2 , a 3 q ∖ ta 4 , u, vu.
We lastly need a Hall-type matching condition for 'bipartite' hypergraphs due to Haxell, which has been used frequently for embedding problems in random graph theory.
Theorem 2.12 (Haxell's condition [30]).Let H be an ℓ-graph whose vertex set can be partitioned into sets A and B such that |e X A| " 1 and |e X B| " ℓ ´1, for every edge e P EpHq.Suppose that for every choice of subsets A 1 Ď A and B 1 Ď B such that |B 1 | ď p2ℓ ´3qp|A 1 | ´1q, there exists an edge e P EpHq intersecting A 1 but not B 1 .Then H has an A-saturating matching (i.e. a collection of disjoint edges whose union contains A).

The sparse regularity method for hypergraphs
Following [20,19] in a natural generalisation of the analogous concept for graphs, we say that, given ε ą 0 and p P p0, 1q, a 3-partite 3-graph where d G pA, B, Cq " e G pA, B, Cq{p|A||B||C|q stands for the density of edges of a given triple.
A partition pV i q iPrts of the vertex set of a 3-graph G is said to be pε, pq-regular if it is an equipartition and for all but at most ε `t 3 ˘triples V i , V j , V k , the graph induced by them is pε, pqregular.For the (hyper)graph regularity lemma to be of any use, it usually needs to prevent too many edges lying within some partition class V i .A common way to restrict this is the notion of upper-uniformity.We say that a 3-graph G is pη, b, pq-upper-uniform, for some η P p0, 1q and With all these concepts at hand, we state a so-called weak 6 hypergraph regularity lemma, which acts as a natural generalisation from the graph setting, can be proven in the same way (see, e.g.[38,41,25] for the sparse regularity lemma and [40] for the regularity lemma in dense hypergraphs), and appears in the same form in [20,19].Theorem 3.1.For every ε ą 0 and t 0 , b ě 1 there exist η ą 0 and T ě t 0 such that for every p P p0, 1s, every pη, b, pq-upper-uniform 3-graph G with at least t 0 vertices admits an pε, pq-regular partition pV i q iPrts , where t 0 ď t ď T .
The upper-uniformity property can be seen as a 'true property of random graphs' and indeed is exhibited by H 3 pn, pq with high probability (e.g.established by a straightforward application of Chernoff's inequality and the union bound).Lemma 3.2.For every η P p0, 1q and b ą 1 the random 3-graph H 3 pn, pq is pη, b, pq-upper-uniform with probability at least 1 ´e´ωpn log nq , provided that p " ωpn ´2 log nq.
Given an equipartition pV i q iPrts of the vertex set of a 3-graph G, we define the reduced graph R " RppV i q iPrts , ε, p, αq on vertex set t1, . . ., tu corresponding to the sets V i , whose edges are all 3-element sets of indices ti, j, ku such that d G pV i , V j , V k q ě αp and GrV i , V j , V k s is pε, pq-regular.
In fact, in the regularity lemma, one can even roughly define where the clusters V i lie in the graph G. Namely, given a partition V pGq " P 1 Y ¨¨¨Y P h , the pε, pq-regular partition resulting from Theorem 3.1 can be made such that all but at most εht clusters V i each completely belong to one P j (may be distinct for different V i ).This comes in handy when it comes to finding absorbers.
The property that we use most frequently is that the reduced graph in a way inherits degree properties from its underlying graph G.This is nothing fancy and should come as no surprise to anyone familiar with the (graph) regularity method.Again, very conveniently, one can make it such that degrees are 'controlled' within certain predetermined sets, and not only globally in the whole graph R. The following statement is almost a one-to-one copy of [20, Lemma 4.7] (slightly paraphrased for convenience).

Lemma 3.3 ([20, Lemma 4.7]
).For all h, t 0 P N and ε, δ, λ ą 0, there exist η, b, T ą 0 such that the following holds.Let p P r0, 1s and let G be a sufficiently large n-vertex pη, b, pq-upper-uniform 3-graph.Let n 1 , . . ., n h ě λn and P 1 , . . ., P h be a partition of V pGq with |P i | " n i .Then there exists an pε, pq-regular partition pV i q iPrts of V pGq, for some t P rt 0 , T s and a corresponding reduced t-vertex 3-graph R " RppV i q iPrts , ε, p, 2εq with the following property.Let P i be the set of clusters V j contained entirely in P i and let t i " |P i |.Then: (i) t i ě p n i n ´εhqt for every i P rhs.(ii) Let d P t1, 2u.Suppose that for some i, j P rhs all but at most opn d q of the d-sets S Ď P i satisfy Then all but at most ?ε `t d ˘of the d-sets S Ď P i satisfy deg R pS, P j q ě pδ ´ph `2qε ´?ε ´3{t 0 q ˆtj ´d 3 ´d ˙.
We remark that, if one wants to only inherit minimum degree (that is, d " 1), then standard double counting methods (see e.g.[57]) show that this can be done without having the ?εt error term.Namely, actually all vertices satisfy the corresponding degree assumption.For d ě 2, the 'almost all' is necessary.
Lastly, we use a hypergraph version of the infamous K LR conjecture7 whose proof for linear hypergraphs was explicitly spelled out in [19] but already observed to hold in the work of Conlon, Gowers, Samotij, and Schacht [12].For a 3-graph H on vertex set t1, . . ., tu we denote by GpH, n, m, p, εq the class of graphs G obtained in the following way.The vertex set of G is a disjoint union V 1 Y ¨¨¨Y V t of sets of size n.For each edge ijk P EpHq we add to G an pε, pq-regular 3-graph with m edges between the triple pV i , V j , V k q (and these are the only edges of G).A canonical copy of H in G is a t-tuple pv 1 , . . ., v t q with v i P V i for every i P V pHq and v i v j v k P EpGq for every ijk P EpHq.We write GpHq for the number of canonical copies of H in G. Lastly, we need the notion of 3-density m 3 pHq of a 3-graph H, which is defined as ) .
Strictly speaking, the subgraph G that we later apply Theorem 3.4 to is not a member of GpH, n, m, p, εq, in particular not all pε, pq-regular 3-graphs have exactly m edges-the number of edges across these are within a constant factor of each other.However, subsampling (see, e.g.[25,Lemma 4.3]) circumvents this.For clarity of presentation, we prefer to not explicitly spell out this argument.

Covering random hypergraphs by loose paths
In this section we show a vital part of every strategy relying on the absorbing method which in our specific problem reads as: the majority of vertices of G Ď H 3 pn, pq can be covered by a few (loose) paths.For the purposes of this lemma, we consider single vertices to be loose paths of length zero.
Lemma 4.1.Let d P t1, 2u.For every γ, ϱ ą 0, there exists a C ą 0 such that w.h.p.H 3 pn, pq has the following property, provided that p ě C log n ¨maxtn ´3{2 , n ´3`d u.Let G Ď H 3 pn, pq with vpGq ě n{2 in which all but opn d q d-sets S Ď V pGq satisfy deg G pSq ě pδ d `γqp `n´d 3´d ˘.Then there exist at most ϱn vertex-disjoint loose paths that cover the vertex set of G.
To a reader familiar with the topic, there is no magic that happens here.We apply the sparse regularity lemma to G, find a desirable structure in the obtained reduced graph, and then use it as a guide to construct loose paths in G itself.Perhaps the most innovative thing comes in the part where, in an pε, pq-regular triple pV 1 , V 2 , V 3 q, we find a long loose path covering all but op|V i |q vertices in each V i .This itself relies on a widely-used Depth-First Search (DFS) technique employed explicitly in the context of graphs in [5,6].(For an extremely neat application of the method and more in-depth discussion see [45].) Then there is a loose uv-path of length 2t ´4k in H with u, v P V 1 Y V 3 , and whose every other degree-one vertex lies in V 2 .
Proof.We explore the graph H by using a variant of the Depth-First Search (DFS) procedure as follows.To start with, set T " ∅, S " ∅, and U " V pHq.For as long as U X pV 1 Y V 3 q ‰ ∅, do: • if S " ∅, pick an arbitrary vertex from U X pV 1 Y V 3 q and add it to S; • if S ‰ ∅ and there is an edge uvw P H such that u P V i is the last added vertex to S, and v P U X V 2 and w P U X V 4´i , move v, w from U to S in that order; • otherwise, for uvw P H being the last three vertices added to S, move v, w from S to T (if there is no such edge just move the only vertex u in S to T ).
Observe that the vertices in S at all times span a loose path as wanted in the lemma, but maybe not of the required length, and, crucially, there is never a time when some edge uvw P H is such that u P T X V i , v P U X V 2 , and w P U X V 4´i , for i P t1, 3u.We aim to show that at some point we have |S X V 2 | ě 2t ´4k `1 which is sufficient for the lemma to hold.
In every step of the procedure either one (if S " ∅, say) or two vertices get moved from U to S or from S to T .Consider the first moment in time when |T X V i | " k, for some i P t1, 3u; we may safely assume this happens for V 1 .So, |T X V 3 | ă k, and moreover by the description of the procedure above, at this point we necessarily have , by the property of the lemma that such three sets must contain an edge.
Since p " ωpn ´2 log nq, by Lemma 3.2 H 3 pn, pq is w.h.p. pη{2, b, pq-upper-uniform, and so G is pη, b, pq-upper-uniform then.Therefore, we can apply Lemma 3.3 to G with δ d `γ (as δ), to obtain an pε, pq-regular partition pV i q iPrts for some t 0 ď t ď T and a corresponding reduced graph R " RppV i q iPrts , ε, p, 2εq.In particular, we get that for all but at most ?ε `t d ˘d-sets S Ď V pRq, it holds that deg R pSq ě pδ d `γ{2q `t´d 3´d ˘.Let us remove from each V i at most two vertices to get that they are all of even size tn{tu ´1 ď m ď rn{ts (with slight abuse of notation we still refer to them as V i ).
Let R 1 , . . ., R t{s be a partition of V pRq into disjoint subsets of size s, chosen uniformly at random.Recall that Λ :" `s d ˘p? ε `e´Ωpγ 2 sq q.By Lemma 2.7, with positive probability all but a Λ-fraction of the subgraphs R i :" RrR i s have minimum d-degree at least pδ d `γ{4q `s´d 3´d ˘.As we have picked s to be even and large, each R i with this minimum degree has a loose Hamilton cycle by Theorem 2.10.The vertices not covered by loose cycles in R are then at most Λt `s.We next show how to cover each loose cycle in R with not too many loose paths in G.
Consider a (loose) Hamilton cycle in some R i and suppose without loss of generality that the clusters corresponding to the vertices of R i are V 1 , . . ., V s in the order in which they appear on the cycle.Then we know that pV i , V i`1 , V i`2 q is pε, pq-regular with density at least 2εp, for every i P rs ´1s, i odd (s `1 is identified with 1).Split every V i , for i odd, arbitrarily into By the definition of a regular triple, for all , each of size εm, we have and in particular e G pX i , X i`1 , X i`2 q ą 0. Lemma 4.2 implies there is a loose uv-path in , which is of length 2 ¨m{2 ´4εm " p1 ´4εqm and thus uses all but at most 8εm vertices in . Repeating this for every odd i P rss, we get s{2 loose paths covering all but at most 8εm ¨s{2 vertices in V 1 , . . ., V s .The whole thing can independently be repeated for every R i as well.In total, and counting vertices as paths of length zero, we found at most t{s ¨s{2 ¨p1 `8εmq `pΛt `sq ¨m `2t ď t ¨8εn{t `2Λn `opn{tq ď ϱn loose paths that cover the vertex set of G.

Connecting Lemma
In this section we prove a vital ingredient both for constructing absorbers and independently as a part of the proof of Theorem 1.1.Roughly speaking, we show that one can connect prescribed pairs of vertices with short paths through a set of vertices under some degree assumptions.Given a set W Ď V pGq in a graph G, an integer ℓ, and a set of pairs tx i , y i u i from V pGq ∖ W , a ptx i , y i u i , W, ℓq-matching is a collection of internally vertex-disjoint paths P i , where each P i is of length at most ℓ, has x i , y i as endpoints, and its remaining vertices belong to W . Lemma 5.1 (Connecting Lemma).Let d P t1, 2u.For every γ, ξ ą 0 there exist ε, C ą 0 such that w.h.p.H 3 pn, pq satisfies the following, provided p ě C log n ¨maxtn ´3{2 , n ´3`d u.Let G Ď H 3 pn, pq and U, W Ď V pGq be disjoint subsets such that Then, for every family of distinct pairs tx i , y i u iPrts in U such that t ď ε|W | and every u P U appears in at most two pairs, there exists a ptx i , y i u i , W, 4q-matching in G.
Proof.Given ξ and γ, let γ, ε ą 0 be sufficiently small and C ą 0 sufficiently large for the arguments below to go through.Condition on H 3 pn, pq having the properties of Lemma 2.2, Lemma 2.3, Lemma 2.4, Lemma 2.5 for γ (as γ), and Lemma 2.6, which happens with high probability.
Consider an auxiliary hypergraph H (an 8-graph in case d " 1 and a 2-graph in case d " 2) with vertex set rts Y W in which for Z Ď W , with |Z| " 7 if d " 1 and |Z| " 1 if d " 2, an edge tiu Y Z exists if and only if there is a x i y i -path in G all whose internal vertices belong to Z. Hence, a rts-saturating matching in H corresponds to a ptx i , y i u i , W, 4q-matching in G. Our plan is to use Haxell's matching theorem (Theorem 2.12) to show this graph contains such a matching.It is sufficient to, for every I Ď rts and Q Ď W of size |Q| ď 16|I|, find an i P I and an x i y i -path in G whose internal vertices all belong to W ∖ Q.Indeed, this implies there is an edge in H that intersects I but does not intersect Q, and so the condition in Theorem 2.12 is satisfied.We treat the cases d " 1 and d " 2 separately, starting with d " 2.
Codegree, d " 2. Recall, δ 2 " 1{4.Consider any I Ď rts and Q Ď W of size |Q| ď 16|I|.Let P be a maximal set of pairs tx i , y i u with i P I and all vertices distinct and assume towards a contradiction there is no x i y i -path of length at most four in G with all internal vertices belonging to W ∖ Q.By the assumption of the lemma we have |P| ě |I|{3.If |P| ď ε ´3 log n{p, using on the one hand the minimum codegree assumption and on the other the property of Lemma 2.
for all i.From the assumptions of the lemma, it is straightforward to show that such a partition exists by making use of the property of Lemma 2.6 and the union bound.
Consider any I Ď rts and Q Ď W of size |Q| ď 16|I|.Assume that I contains indices i so that the collection tx i , y i u consists of distinct vertices (a maximal subset of such i comprises at least a third of the original I which has no influence on the proof).The x i y i -path we are trying to find is going to be such that it intersects each W 1 , W 3 in exactly two vertices, and W 2 in exactly three vertices.
Let Wi :" W i ∖ Q.It is sufficient to show that there is an index i P I for which: (B1) There exist S 1 Ď N G px i , W1 q and S 3 Ď N G py i , W3 q, both of size ?n; (B2) Let G 1 and G 3 be auxiliary 2-graphs on the same vertex set W2 and uv P EpG i q if tu, vu form an edge in G with a vertex from S i , i P t1, 3u.Then for each i P t1, 3u Before proving these statements, let us show how to complete the proof.If we were to find three vertices u, v, w P W2 so that e G pS 1 , u, vq ą 0 and e G pS 3 , w, vq ą 0, this would close a x i y i -path as desired.A path of length two in G 1 Y G 3 with one edge in each of G 1 , G 3 corresponds exactly to a triple u, v, w P W2 as above.By (B2) and the pigeonhole principle, there must be a v P W2 with deg G 1 pvq, deg G 3 pvq ě 2. This implies we can choose u as one of v's neighbours in G 1 and w as one of v's neighbours in G 3 such that u ‰ w.
We first show there is an i P I for which (B1) holds.For this, it is sufficient to show that more than half of the indices i P I are such that deg G px i , W1 q ě γp `n´1 2 ˘.The existence of i P I and the corresponding sets S 1 and S 3 as in (B1) follows from the pigeonhole principle and the property of Lemma 2.5.
Let X be a set of vertices x i , i P I, with deg G px i , W1 q ă γp `n´1 2 ˘and suppose |X| ě |I|{2.Owing to the assumption on the minimum degree (1) By ( 1) and for ε sufficiently small, we thus get On the other hand, noting that |P 1 | ď n, by the property of Lemma 2.3 and taking supersets of and so by the property of Lemma 2.4 again ˙¯`e G pS 1 , P 1 q ď p1 `εq|S 1 |p ˆ?7s{4 2 ˙`op|S 1 |ps 2 q, which is smaller than |S 1 |p7{16 `γ{4qp `s 2 ˘by our choice of ε, thereby contradicting (4).
In principle, the idea behind it is simple.It relies on reducing the problem of finding a spanning subgraph S in some graph G to the one of finding an almost spanning subgraph S 1 Ď S, say, one of size p1 ´op1qq|V pGq|.Often times, the latter is significantly easier to solve, if nothing else, just for the fact that we have quite a bit of room for error.In practice, the implementation of this idea typically depends on first embedding a highly structured graph A into G, which is capable of extending any partial embedding to a complete one.The task of designing the graph A with this magical property is where the whole art of absorption lies in.Usually, it is specific to the embedding problem at hand and is where the main difficulties arise-it can be, first, quite surprising that such a graph should even exist and, second, challenging to find it in the host graph G in a convenient way (or in any way for that matter).
When dealing with paths, the 'design' of the graph A is not too complex.We make this more rigorous.
Definition 6.1 (pa, b, Rq-absorber).An pa, b, Rq-absorber is a graph A with R Ď V pAq, a, b P V pAq ∖ R, with the property that for every The next lemma handles the second step of the method: actually finding the graph A in the subgraph G of the random graph H 3 pn, pq.
Let G Ď H 3 pn, pq with δ d pGq ě pδ d `γqp `n´d 3´d ˘.Then, for a uniform random set R Ď V pGq of size αn, w.h.p. there exists an pa, b, Rq-absorber in G of order at most ξn.
It may seem awkward that there are two probabilistic statements in the lemma.Indeed, the first w.h.p. establishes some typical properties a random 3-uniform hypergraph has (such as, e.g., distribution of the edges), whereas the second w.h.p. is over the choice of R. Namely, once we condition on H 3 pn, pq having these 'nice' properties, then for a randomly chosen set there is an absorber with high probability.In fact, the lemma could be written in a quantitative form, saying how w.h.p. in H 3 pn, pq there are at least, say, a p1 ´n´5 q-fraction of choices for R which yield an absorber.We found this a bit more cumbersome to deal with and decided to go with the former.
As per usual, the 'construction' of such a graph A consists of carefully patching up many small structures.

Definition 6.3 (xy-absorber
).An xy-absorber is a graph that consists of: • a path P , which we refer to as the covering path, and • a path Q, which we refer to as the non-covering path, with V pQq " V pP q ∖ tx, yu and whose endpoints are identical with those of P .
We now define the xy-absorber we use, depending on d (in doing this we draw inspiration from [10]).Both of these, however, have a much more natural visual representation depicted in Figure 2 and Figure 1.
Definition 6.4.The xy-absorber A d xy is defined as: (d " 1) consists of: • a cycle x 1 , x 2 , x 3 , x, x 4 , . . ., x 10 , P x , x 1 , where P x is an x 1 x 10 -path of length four; • a cycle y 1 , y 2 , y 3 , y, y 4 , . . ., y 10 , P y , y 1 , where P y is a y 1 y 10 -path of length four; • a copy of A 2 • an x 5 y 5 -path and a v 7 b 1 -path, both of length four.For future reference, the graph obtained by removing P x , P y , the x 5 y 5 -path, and the v 7 b 1 -path is called a backbone of the xy-absorber A 1 xy (see Figure 3a below).We refer to all vertices of A d xy other than x, y as its internal vertices.
To build an pa, b, Rq-absorber, we plan to string together a number of copies of A d xy .Clearly, we cannot just build disjoint xy-absorbers for every pair of vertices x, y P R for the simple reason of there not being enough space for that as `|R| 2 ˘" n.A way of dealing with this originated in the work of Montgomery [53], who used an idea of looking at an auxiliary bounded degree graph, which serves as a template for which pairs of vertices of R to use.Roughly speaking, there is a graph T " pR, Eq on the vertex set R and with ∆pT q " Op1q, so that if for every xy P EpT q we find disjoint xy-absorbers, then the obtained graph is an pa, b, Rq-absorber, for some a, b R R.
The following lemma is very similar to [20,Lemma 7.3], but it has the additional property that the template graph it describes has bounded maximum degree.The proof, being almost identical to that of the mentioned lemma, is for completeness spelled out in the appendix.Lemma 6.5.There is an L ą 0 such that the following holds.For any sufficiently large m, there exists a graph T with 2m ď vpT q ď Lm, maximum degree at most L, and a set Z of m vertices, such that for every Z 1 Ď Z with |Z 1 | ă m{2 and |V pT q ∖ Z 1 | even, the graph T ´Z1 has a perfect matching.
At this point, we essentially reduced our goal to finding a single xy-absorber for a prescribed pair of vertices x, y P R.This is the key lemma of this section.Lemma 6.6.Let d P t1, 2u.For any γ ą 0 there exist λ, C ą 0, such that w.h.p.H 3 pn, pq satisfies the following, provided that p ě C log n ¨maxtn ´3{2 , n We remark that the degree assumption for x, y is not needed in the d " 2 case.The proof of this lemma is the most intricate part of the argument so we defer it to Section 6.1.With all these ingredients we can now prove the absorbing lemma.
Having chosen these, let α be sufficiently small with respect to all constants (in particular, small with respect to the given ξ ą 0) and C sufficiently large.Write m :" αn, let T be the auxiliary graph given by Lemma 6.5 for m, and let Z be the set of special vertices in T as in the lemma.Assume H 3 pn, pq is such that it satisfies the conclusion of Lemma 2.1 with γ{64 (as γ) and 8DL (as C), Lemma 2.6, Lemma 5.1, and Lemma 6.6 with γ{2 (as γ).
Choose a uniform random set R Ď V pGq of size αn and let R 0 Y V Y W be a partition of V pGq ∖ R such that: (C1) |R| " m, |R 0 | " vpT q ´m, and |W | " n{100; (C2) deg G pS, Xq ě pδ d `γ{2qp `|X|´d 3´d ˘, for every d-set S Ď V pGq and X P tR Y R 0 , V, W u. Note that, in particular, |V | " p1 ´1{100 ´op1qqn ě n{2.By the property of Lemma 2.6 almost any choice of such a partition will do.
Let f : V pT q Ñ R 0 Y R be a bijection which maps the vertices belonging to the set Z in T to R. For each edge xy P EpT q, we plan to first find an f pxqf pyq-absorber A d f pxqf pyq whose internal vertices belong to GrV s in such a way that all these absorbers are pairwise internally vertex-disjoint.Subsequently, we string all the f pxqf pyq-absorbers together by an application of the Connecting Lemma (Lemma 5.1) over the set W .
The first part of this is to be completed using Haxell's condition and Lemma 6.6.Let H be a pD ´1q-uniform hypergraph with vertex set U Y V , where U " ␣ tf pxq, f pyqu : xy P EpT q ( .For tu x , u y u P U and S Ď V with |S| " D ´2, we add the edge tu x , u y u Y S to H if and only if there is a copy of A d uxuy in Grtu x , u y u Y Ss.Then what we are looking for is precisely a U -saturating matching in H. Comparing to Haxell's condition (Lemma 2.12), it is sufficient to show that for every , there is some tu x , u y u P U 1 and a copy of A d uxuy in G whose internal vertices are fully contained in V ∖ V 1 .
As ∆pT q ď L, we can greedily find a set of pairwise vertex-disjoint edges xy P EpT q for which tf pxq, f pyqu P U 1 and which is of size at least |U 1 |{p2Lq.For simplicity, we assume U 1 already consists only of such disjoint edges-this changes nothing in the proof.In case d " 2, straightforwardly applying Lemma 6.6 with any f pxq, f pyq for which tf pxq, f pyqu P U 1 (as x, y), GrV Y tf pxq, f pyqus (as G), and V 1 (as S), we are done.Note that we can indeed do so since In the other case, d " 1, the only thing remaining is to check that there are f pxq, f pyq with tf pxq, f pyqu P U 1 that satisfy the degree requirement deg 2 ˘.Towards contradiction, suppose there is no such f pxq, f pyq.Let Q Ď V pGq be the union of all f pxq that violate the prior requirement and such that for some f pyq, tf pxq, f pyqu P U 1 , and assume |Q| ě |U 1 |{2.On the one hand, this means while on the other, by the property of Lemma 2.
which is a contradiction.So, |Q| ă |U 1 |{2 and by the pigeonhole principle, there has to exist a pair tf pxq, f pyqu P U 1 with the desired degree value into Finally, we use Lemma 5.1 to connect all f pxqf pyq-absorbers into one loose path.Denote the absorbers we have found as A 1 , . . ., A t , for t " epT q, each A i a copy of A d xy .We want to, for every i P rt ´1s, connect v 1 P V pA i q and a 1 P V pA i`1 q if d " 1 (see Figure 2) and v 7 P V pA i q and v 1 P V pA i`1 q if d " 2 (see Figure 1).Let Y Ď V be the set of (the images of) all these 2pt ´1q vertices we would like to connect.We use the property of Lemma 5.1 with W , Y (as U ) which we can do by (C2) and since |W | ě n{100 and the total number of pairs we want to connect with paths is epT q ´1 ď L 2 m " L 2 αn ď εn{100 ď ε|W |.
Lastly, observe that the total number of vertices used by this whole procedure is epT q ¨D `pepT q ´1q ¨7 ď L 2 αn ¨pD `7q ď ξn, as desired.
It remains to establish that the graph A obtained by connecting the absorbers A 1 , . . ., A t as described comprises an pa, b, Rq-absorber, where a and b are (the images of) a 1 P V pA 1 q and v 1 P V pA t q if d " 1 and (the images of) v 1 P V pA 1 q and v 7 P V pA t q if d " 2. Namely, let R 1 Ď R of size |R 1 | ă |R|{2 be such that V pAq ∖ R 1 has odd cardinality.Note that V pAq ∖ pR 0 Y Rq must also be of odd cardinality as it can be covered by a loose path (by taking the non-covering paths of all individual f pxqf pyq-absorbers) and hence, crucially, R 0 Y pR ∖ R 1 q is of even cardinality.In particular, this means there is a perfect matching for f ´1pR 0 Y pR ∖ R 1 qq in T .Consider the set of f pxqf pyq-absorbers corresponding to the edges in this matching.Then, as a witness for the absorbing property of A, we can use the covering path for all these absorbers, the non-covering path for all other f pxqf pyq-absorbers, and the short paths connecting the absorbers to get the desired loose path between a and b.

Finding an xy-absorber robustly
In this section we prove Lemma 6.6.The case d " 2 is much simpler, so we deal with it first.
We use the property of Lemma 2.9 to get a set T Ď `V pGq∖S (by choosing C sufficiently large) such that all pairs tu, vu P `V pGq∖S 2 ˘∖ T satisfy deg F pu, vq ě p1{4 `γ{2qppn ´|S| ´2q " Ωplog nq, for F :" G ´S. Recall, our goal is to find a copy of A 2 xy in F (see Figure 1).
Since T is relatively small, there must be some v 4 P V pF q ∖ tx, yu such that neither tx, v 4 u nor ty, v 4 u belong to T .Having chosen v 4 , take distinct v 2 , v 6 P V pF q ∖ tx, y, v 4 u such that v 2 xv 4 , v 4 yv 6 are edges in F .Let V 3 :" tv P V pF q ∖ tx, y, v 2 , v 4 , v 6 u : tv, v 2 u, tv, v 4 u R T u and V 5 :" tx, y, v 2 , v 4 , v 6 u Y tv P V pF q : tv, v 6 u P T u, and note that |V 3 | ě p1 ´10 ´2qn and |V 5 | ď 10 ´5n.It is enough to show that there is an edge v 3 v 4 v 5 in F such that v 3 P V 3 and v 5 P V pF q ∖ V 5 , because tv 2 , v 3 u, tv 5 , v 6 u R T imply that we can choose v 1 and v 7 as desired for A 2 xy .Suppose for contradiction such an edge does not exist in F .Then, from the codegree assumption and the property of Lemma 2.4, we have |V 3 |{2 ¨p1{4 `γ{2qppn ´|S| ´2q ď e F pv 4 , V 3 , V 5 q ď e G pv 4 , V 3 , V 5 q ď 2 ¨10 ´5|V 3 |np, which is a contradiction.Thus, an edge v 3 v 4 v 5 P F exists as desired.
In what follows we provide a proof of Lemma 6.6 for d " 1.This is the most involved part of the whole proof and we, for convenience of reading, first give a brief outline.The main idea relies on an intricate combination of the sparse regularity method and the connecting lemma.It consists of two almost independent steps: (1) find a backbone of an A 1 xy absorber; (2) use the Connecting Lemma (Lemma 5.1) to find the remaining loose paths which comprise an absorber A 1 xy (see Figure 2).Most of the difficulty lies in the first part.To do this, we use the 'contraction technique' of Ferber and Kwan [20] and the regularity method.In order for the next steps to make more sense, we first introduce a definition.Definition 6.7.A contracted backbone of an absorber is a graph that consists of: • edges x 1 x 7 x 8 , x 8 x 9 x 10 and a 1 a 2 a 3 , a 2 a 4 x 1 , a 3 a 4 x 9 ; • edges y 1 y 7 y 8 , y • a copy of A 2 x 7 y 7 .
A contracted backbone of an absorber can be thought of as starting with a backbone of A 1 xy and contracting the edges x 1 x 2 x 3 , x 3 xx 4 , x 4 x 5 x 6 into a single vertex x 1 , and keeping only the edges 'to the outside' (that is, ones containing x 2 or x 6 ).The same is done to obtain y 1 .Note that, in this context, the vertices x 1 and y 1 play a special role and we often explicitly mention them when talking about the contracted backbone of an absorber.Strictly speaking, a name for this structure that is more descriptive of the aforementioned contraction operation might be a 'contracted backbone of an xy-absorber'.However, as the graph itself does not contain the vertices x, y, we omit them from the name to avoid confusion.For a more natural visual representation we depict this contraction operation below.The contraction operation on G Ď H 3 pn, pq, almost analogous to the one of Ferber and Kwan, is defined as follows.For a 3-graph G, a collection U of disjoint sets U 1 , U 2 Ď V pGq, and a family of disjoint 4-tuples F Ď V pGq we let GpU, Fq be a 3-graph on vertex set U 1 Y U 2 Y F and whose edge set is given as follows: add first all edges from GrU 1 Y U 2 s and next, for every w " pw 1 , w 2 , w 3 , w 4 q P F, we add an edge wuv to GpU, Fq if and only if u, v P U 1 and w 2 uv P EpGq or u, v P U 2 and w 4 uv P EpGq.
We can now continue with the outline.Namely, imagine for a moment that we can find a collection of Ωpnq distinct 4-tuples px 1 , x 2 , x 5 , x 6 q such that, x 1 x 2 x 3 , x 3 xx 4 , x 4 x 5 x 6 P EpGq, for some x 3 , x 4 .Let us, for every such tuple, contract these vertices (and edges) into a single vertex and keep only edges to the outside that contain x 2 or x 6 (as in the contracted graph Gp¨, ¨q above).Denote the set of these new vertices as X 1 .Now, do the same for y while keeping all these disjoint to obtain Y 1 .If we were to find a copy of the contracted backbone of an absorber with x 1 , y 1 mapped into some vertices of X 1 and Y 1 respectively, we would be done by just undoing the contraction operation.
To do this last part, we rely on the sparse regularity lemma (Lemma 3.3) and Theorem 3.4.If all the previous steps have been done carefully, what remains of the graph still satisfies all the necessary conditions (in particular, the minimum degree will be sufficiently large) to do this.First, we show that in the reduced graph obtained from the application of Lemma 3.3 we can find a copy of the contracted backbone of an absorber with the vertices x 1 and y 1 being mapped to their corresponding 'clusters' belonging to X 1 and Y 1 .Subsequently we use Theorem 3.4 to transfer it into a canonical copy of the same graph in G8 .As we prove in the appendix (not to interrupt the flow of the main argument), the contracted backbone of an absorber is just sparse enough to exist in the regular partition.Claim 6.8.The m 3 density of a contracted backbone of an absorber is 2{3.
There is one crucial difference compared to the method used in [20], reflected in the fact that we work with p " Θpn ´3{2 log nq, in comparison to p " Ωpn ´1q.Namely, the neighbourhood of a vertex in our setting is roughly of size n 2 p " Θp ?n log nq, which is much below the point at which we can rely on regularity properties.Hence, we first need to show using ad-hoc density techniques that x and y expand to Θpnq vertices in two hops, and from then on start implementing the strategy outlined above.This all also affects the design of our xy-absorbers.
As a final preparation step, we need a statement for dense hypergraphs that enables us to find a copy of the contracted backbone of an absorber in the reduced graph.Lemma 6.9 (Proposition 8 in [10]).For every γ P p0, 3{8q the following holds for every sufficiently large n.Suppose H is a 3-uniform hypergraph on n vertices which satisfies δ 1 pHq ě p5{8 `γq 2  `n 2 ˘.
Then for every pair of vertices x, y P V pHq the number of 7-tuples that form a copy of A 2 xy is at least pγnq 7 {8.
Proof of Lemma 6.6, case d " 1. Recall, δ 1 " 7{16.Given γ, let t 0 P N be a large constant, in particular such that 3{t 0 !γ.Next, let ξ " ξ 2.5 pγ{256q, λ " λ 2.8 pγq, choose ε sufficiently small, and let η and b be as given by Lemma 3.3 for 7{16 `γ{8 (as δ) and other respective parameters.Pick µ sufficiently small with respect to ε, t 0 , and ξ.Let G 1 :" G ´S.We condition on H 3 pn, pq satisfying the conclusion of Lemma 2.5, Lemma 2.6, Lemma 2.8, and Lemma 5.1.Furthermore, (D1) for all disjoint U 1 , U 2 Ď V pGq and F Ď V pGq 4 , with |U i |, |F| ě ξn, the conclusion of Theorem 3.4 holds for GpU, Fq with U pU 1 , U 2 q: for every H with m 3 pHq ď 2{3, every subgraph G 0 of GpU, Fq which belongs to GpH, µn, 2εpµ 3 n 3 , p, εq contains a canonical copy of H; (D2) every GpU, Fq as in (D1) is pη, b, pq-upper-uniform; Let us establish (D1) and (D2).Observe that GpU, Fq has Ωpnq vertices and that it can be coupled with a random graph on its vertex set with edge probability p. Namely, there is a bijection φ from edges of GpU, Fq to 3-sets in V pGq such that the existence of e as an edge in GpU, Fq is determined by φpeq being an edge in G or not.
So, for a fixed choice of U 1 , U 2 , and F, the graph GpU, Fq satisfies both (D1) and (D2) with probability at least 1 ´e´Ωpn 3 pq ´e´ωpn log nq .As there are at most 2 2n ¨n4n ď e 10n log n choices for these sets, recalling that p ě Cn ´3{2 log n, a simple union bound shows that w.h.p. both (D1) and (D2) hold in H 3 pn, pq.Thus, from now on we also condition on (D1)-(D2).
By the property of Lemma 2.8 there is a set T Ď V pGq ∖ tx, yu with |T | ď ?n such that the graph G 2 :" G ´pS Y T q has minimum degree at least p7{16 `γ{4qp `|V pG 2 q|´1 2 ˘.Hence, for simplicity of notation, we assume that G 1 already satisfies this.
2 ˘for every v P V pG 1 q and Z P tV x , V y , U 1 , U 2 , W u. Observe that |Z| ě n{6.A vast number of partitions is such by the property of Lemma 2.6, so we fix one of them.In order to complete the proof it is sufficient to show that (i ) there exists a backbone of ) for a copy of a backbone of A 1 xy as above, there is an x 1 x 10 -path, a y 1 y 10 -path, a x 5 y 5 -path, and a v 7 b 1 -path, each of length four, and whose internal vertices all belong to W .
Throughout the proof and for ease of reference, it might help to have Figure 2 and especially Figure 3 in mind.Assuming we have found the backbone of A 1 xy , step (ii ) follows by applying the Connecting Lemma with W and (the images of) ttx 1 , x 10 u, ty 1 , y 10 u, tx 5 , y 5 u, tv 7 , b 1 uu as the family of pairs to be connected by paths.For the rest of the proof, we focus on showing (i ).
We can now apply Lemma 6.9 with R 2 :" R 1 rU 2 ∖ tX 8 , X 9 , X 10 , Y 8 , Y 9 , Y 10 us (as H) and X 7 , Y 7 (as x, y), since ḟor some small γ 1 ą 0. (Note that here we used (E1) and that t can be assumed to be sufficiently large.)We conclude that there is a copy of A2 X 7 Y 7 in R 2 .Next, we apply Lemma 2.11 with RrU 1 Y tX 1 , X 9 us (as H) and X 1 , X 9 (as u, v) to find vertices Almost analogously, there are B 1 , B 2 , B 3 , B 4 P U 1 so that tB 1 , B 2 , B 3 u, tB 2 , B 4 , Y 1 u, tB 3 , B 4 , Y 9 u P EpR 1 q.This forms the contracted backbone of an absorber in R 1 Ď R (see Figure 3b).
As the 3-density of the contracted backbone of an absorber is at most 2{3 by Claim 6.8, we can use (D1) in the subgraph of J induced by the clusters corresponding to the found contracted backbone of an absorber in R 1 .This gives us a canonical copy of the contracted backbone of an absorber in J with vertices x 1 and y 1 mapped into X 1 and Y 1 as desired, and this finally completes the proof.
7 Putting everything together: Proof of Theorem 1.1 With all the preparatory lemmas at hand, the proof of our main theorem follows the usual steps: (i) find an appropriate set R and a not-too-large pa, b, Rq-absorber in G; (ii) cover almost everything else by opnq loose paths; (iii) use the Connecting Lemma to patch those paths together over the set R; (iv) absorb the unused vertices of R into a loose Hamilton cycle.
Pick a random set of vertices R Ď V pGq of size |R| " αn.By the Absorbing Lemma (Lemma 6.2) we get, w.h.p. over the choice of R, an pa, b, Rq-absorber A, for some a, b P V pGq ∖ R, of size at most λn.Furthermore, by Lemma 2.6, w.h.p. we have that every d-set S Ď V pGq satisfies deg G pS, Rq ě pδ d `γ{2qp `|R|´d 3´d ˘.In particular, as H 3 pn, pq has the property of the Connecting Lemma (Lemma 5.1), the set R can be used as the 'reservoir' (set W in the lemma) to find paths through it.We fix such a choice of R and A for the remainder of the proof.
For step (ii) of the strategy, we aim to cover almost all of G ´V pAq with a few loose paths, using Lemma 4.1.If d " 1 then by Lemma 2.8 for V pAq (as S) there exists a set T of size ?n such that F :" G ´pV pAq Y T q satisfies δ 1 pF q ě p7{16 `γ{4qp `|V pF q|´1 and by Lemma 2.9 again for V pAq (as S), in F :" G ´V pAq, all but at most Oplog n{pq " opn 2 q pairs of vertices u, v P V pF q have deg F pu, vq ě p1{4 `γ{2qppn ´|V pAq| ´2q.Thus, in both cases the graph F satisfies the requirements of Lemma 4.1 and, therefore, there are k ď ϱn disjoint loose paths that cover its vertices.Denote the i-th such path by P i and let x i , y i be its endpoints (note, for some i we may have x i " y i ).
Step (iii) is to use the Connecting Lemma (Lemma 5.1) to connect everything into a large loose cycle.Let T " tv 1 , . . ., v t u (note, t " 0 if d " 2).We want to apply it with pairs: ty i , x i`1 u iPrk´1s Y ty k , v 1 u Y tv i , v i`1 u iPrt´1s Y tv t , au Y tb, x 1 u.
Recall, set R is chosen so that |R| " αn and each d-set Q Ď V pGq has deg G pQ, Rq ě pδ d γ{2qp `|R|´d 3´d ˘.Furthermore, the total number of pairs to connect is at most k`t`1 ď ?n`ϱn`1 ď ε|R|.Therefore, by Lemma 5.1 all these pairs can be connected via disjoint loose paths, each of length at most 4, whose internal vertices belong to R. These paths use a subset R 1 Ď R with |R 1 | ď 100ϱn ă αn{2 " |R|{2.The union of these paths with P 1 , . . ., P k makes up a loose ab-path P ˚with vertex set pV pGq ∖ V pAqq Y ta, bu Y R 1 .Notice that this implies n ´|V pAq| `|R 1 | is odd, and since n is even, |V pAq| ´|R 1 | must be odd as well.Lastly, we use the absorbing property of A to find a loose ab-path P A with V pP A q " V pAq ∖ R 1 .The union of P ˚and P A gives us a loose Hamilton cycle in G as desired.

A Complementary proofs
Proof of Lemma 6.5.The proof is almost identical to the proof of [20,Lemma 7.3] with k " 2. We also make use of the following result from [53].
Lemma A.1 ([53, Lemma 10.7]).For any sufficiently large s, there exists a bipartite graph R with vertex parts X and Y Y Z, where Y and Z are disjoint, with |X| " 3s and |Y | " |Z| " 2s, and maximum degree 100, such that if we remove any s vertices from Z, the resulting bipartite graph has a perfect matching.
We construct T by starting with the bipartite graph R from Lemma A.1 with s " rm{2s and potentially deleting one vertex from Z to ensure it has size m.Let G be an m-vertex graph with maximum degree 4 and no independent set of size m{2 (e.g.take the square of a cycle on m vertices) and add to T the edges of G, placed on the vertex set Z. Thus, T has at most 7s ď 4m vertices, and its maximum degree is at most 100 `4 " 104.
Consider a set Z 1 Ď Z with |Z 1 | ă m{2 such that V pT q ∖ Z 1 has even cardinality.Since G has no independent set of size s, we can construct a matching M 1 in T rZ ∖ Z 1 s by repeatedly taking away edges one by one until precisely s vertices are left.These remaining s vertices of Z along with X and Y induce a subgraph in T which contains a perfect matching M 2 by Lemma A.1.Therefore, M 1 Y M 2 is a perfect matching of T ´Z1 .
Proof of Claim 6.8.We first prove a claim that allows us to split the contracted backbone of an absorber into three subgraphs and consider each of them separately.
Claim A.2. Let H be a 3-graph consisting of two linear 3-graphs H 1 and H 2 intersecting in a single vertex.Then if α ě 1{2 and m 3 pH i q ď α for i P t1, 2u, we have m 3 pHq ď α.
Consider the components of the contracted backbone of an absorber obtained after removing the edges of A 2 x 7 y 7 (see Figure 3b).Denote the component which contains x 1 , respectively y 1 , by S x 1 , respectively S y 1 .By Claim A.2, it is enough to show that the 3-density of A 2 x 7 y 7 and of S x 1 are each at most 2{3.
To do this, consider a 3-graph H that is a subgraph of either A 2 x 7 y 7 or S x 1 .It suffices to consider only subgraphs with no isolated vertices and with epHq ě 2. If epHq " 2, we have vpHq ě 5 since A 2 x 7 y 7 and S x 1 are both linear hypergraphs, so epHq´1 vpHq´3 ď 2 3 .If epHq " 3, then vpHq ě 6 since the union of any two edges contains at least 5 vertices, and an extra edge requires an extra vertex (otherwise it must have at least two vertices in common with at least one of the first two edges).Thus, epHq´1 vpHq´3 ď 2 3 in this case as well.If epHq " 4, we have vpHq ě 8 since A 2 x 7 y 7 and S x 1 each have 9 vertices and 5 edges, and in each of them there is no edge with two vertices of degree 1.Thus, epHq´1 vpHq´3 ď 3 5 ă 2 3 again.Finally, epHq " 5 implies H is either A 2 x 7 y 7 or S x 1 and so vpHq " 9 and epHq´1 vpHq´3 " 4 6 " 2 3 .

´2q m ˙pm ď ´2eabpn ´2q γappn ´1qpn ´2q ¯mp m ď ´2eb γpn ´1q ¯m ď ´4eλC γ ¯m.
Set m :" γap `n´1 2 ˘.Fix a ď λn, b ď Ca, and two disjoint sets A and B of sizes a and b, respectively.The probability that e G pA, B, V pGqq ě m is at most ˆabpn

Figure 2 :
Figure 2: The xy-absorber A 1 xy .The dashed lines represent paths of length four.

8 y 9 y
10 and b 1 b 2 b 3 , b 2 b 4 y 1 , b 3 b 4 y 9 ; The contracted of an absorber A 1 xy .

Figure 3 :
Figure 3: The backbone and contracted backbone of an absorber A 1 xy .

´3 log n{pnpq by our choice of p for C ą 0 large enough. From the minimum degree assumption on the one hand, and the property of Lemma 2.2 (ii ) (considering a superset of txu of size ε ´3 log n{pnpq if necessary) on the other, we have
2 ˘, there exist F x Ď pW ∖txuq 4 and P x Ď `W ∖txu EpGq, for every tu, vu P P x , and (A2) for every pw 1 , w 2 , w 3 , w 4 q P F x there is some tu, vu P P x for which uw 1 w 2 , vw 3 w 4 P EpGq.
´3`d u.Let G Ď H 3 pn, pq with δ d pGq ě pδ d `γqp `n´d 3´d ˘and let S Ď V pGq with |S| ď λn.For any x, y P V pGq ∖ S, that if d " 1 additionally satisfy deg G´S pxq, deg G´S pyq ě pδ d `γ{2qp `n´1 2 ˘, there exists a copy of A d xy in G ´S.