The extremal number of longer subdivisions

For a multigraph $F$, the $k$-subdivision of $F$ is the graph obtained by replacing the edges of $F$ with pairwise internally vertex-disjoint paths of length $k+1$. Conlon and Lee conjectured that if $k$ is even, then the $(k-1)$-subdivision of any multigraph has extremal number $O(n^{1+\frac{1}{k}})$, and moreover, that for any simple graph $F$ there exists $\varepsilon>0$ such that the $(k-1)$-subdivision of $F$ has extremal number $O(n^{1+\frac{1}{k}-\varepsilon})$. In this paper, we prove both conjectures.


Introduction
For a multigraph F , a subdivision of F is a graph obtained by replacing the edges of F with pairwise internally vertex-disjoint paths of arbitrary lengths. The k-subdivision of F is the graph obtained by replacing the edges of F with pairwise internally vertex-disjoint paths of length k + 1, and is denoted by F k .
Many researchers have studied the problem of estimating the number of edges needed in a graph G on n vertices to guarantee that it contains as a subgraph a subdivided copy of a fixed graph. The first result in this direction is due to Mader [12] who proved that for any graph F there exists a constant c F = c such that if an n-vertex graph G contains at least cn edges, then G contains a subdivision of F as a subgraph. In this result the size of the subdivided graph can grow with n, which is necessary since an n-vertex graph with cn edges need not contain a cycle of bounded length.
Answering a question of Erdős about planar subgraphs [5], Kostochka and Pyber [11] proved that any n-vertex graph with at least 4 t 2 n 1+ε edges contains a subdivided K t with at most 7t 2 log t ε vertices. This is the first result that guarantees a subdivided K t of bounded size.
For a family F of graphs, we let ex(n, F) be the maximum number of edges in an n-vertex graph not containing any F ∈ F as a subgraph. When F = {F }, we write ex(n, F ) for the same function.
Let F t,k be the family of graphs that can be obtained by replacing the edges of K t with pairwise internally vertex-disjoint paths of length at most k. Jiang [9] proved that for any t ∈ N and any 0 < ε < 1/2, we have ex(n, F t,⌈10/ε⌉ ) = O(n 1+ε ). Here the asymptotic notation means that n → ∞ and other parameters are constant. We follow the same convention throughout the paper.
Note that Jiang's result improves that of Kostochka and Pyber in two ways. Firstly, any F ∈ F t,⌈10/ε⌉ has at most ct 2 ε vertices, so a log factor is saved. Secondly, the edges in Jiang's theorem are replaced by uniformly short paths not depending on t. However, they can still have different lengths. The next result of Jiang and Seiver guarantees a subdivided K t with prescribed path lengths. Theorem 1.1 (Jiang-Seiver [10]). For any t ∈ N and any even k ∈ N, ex(n, K k−1 t ) = O(n 1+ 16 k ).
Note that if k is odd, then K k−1 t is not a bipartite graph, so ex(n, K k−1 t ) = Θ(n 2 ). Conlon and Lee conjectured that the following two strengthenings hold. [4]). Let F be a multigraph and let k ≥ 2 be even. Then ex(n, F k−1 ) = O(n 1+ 1 k ). [4]). Let F be a simple graph and let k ≥ 2 be even. Then there exists some ε > 0 such that

Conjecture 1.2 (Conlon-Lee
In the case k = 2, Conjecture 1.2 follows from the r = 2 case of a result of Füredi [7] and Alon, Krivelevich and Sudakov [1], which states that any bipartite graph with maximum degree at most r on one side has extremal number O(n 2−1/r ). The k = 2 case of Conjecture 1.3 was proved by Conlon and Lee [4], and improved bounds were given by the author [8].
Then there exists some ε > 0 such that As a simple corollary, they significantly improved the bound in Theorem 1.1.
Theorem 1.5 (Conlon-Janzer-Lee [3]). Let F be a simple graph and let k ≥ 2 be even. Then there exists some ε > 0 such that In this paper, we prove both Conjecture 1.2 and Conjecture 1.3.
Theorem 1.6. Let F be a multigraph and let k ≥ 2 be even. Then Theorem 1.7. Let F be a simple graph and let k ≥ 2 be even. Then there exists some ε > 0 such that Note that these results are tight. Indeed, by a result of Conlon [2], the Theta graph θ k,ℓ has extremal number Θ(n 1+1/k ) for all ℓ ≥ ℓ 0 (k), showing that Theorem 1.6 is tight. Moreover, Erdős-Rényi random graphs show that ex(n, K k−1 t ) = Ω(n 1+1/k−c k,t ) where c k,t → 0 as t → ∞, so Theorem 1.7 is also tight.
The rest of the paper is organised as follows. In Section 2, we introduce some of the key definitions and give the high-level structure of the proof, with the key technical lemmas deferred to Sections 3 and 4.

The high-level structure of the proof
is the degree of vertex v. The following lemma, which is a small modification of a result proved by Erdős and Simonovits [6], allows us to restrict our attention to almost regular host graphs. Lemma 2.1 (Jiang-Seiver [10]). Let ε, c be positive reals, where ε < 1 and c ≥ 1. Let n be a positive integer that is sufficiently large as a function of ε. Let G be a graph on n vertices with e(G) ≥ cn 1+ε . Then G contains a K-almost-regular subgraph G ′ on m ≥ n ε−ε 2 2+2ε vertices such that e(G ′ ) ≥ 2c 5 m 1+ε and K = 20 · 2 1 ε 2 +1 .
Using this lemma, Theorem 1.6 and Theorem 1.7 reduce to the following two statements, respectively. For notational convenience, we have dropped the assumption that k is even, and replaced k by 2k.
Theorem 2.2. Let F be a multigraph and let k ≥ 1. Suppose that G is a K-almost-regular graph on n vertices with minimum degree δ = ω(n 1 2k ). Then, for n sufficiently large, G contains a copy of F 2k−1 .
Theorem 2.3. Let F be a simple graph and let k ≥ 1. Then there exists ε > 0 with the following property. Suppose that G is a K-almost-regular graph on n vertices with minimum degree δ = ω(n 1 2k −ε ). Then, for n sufficiently large, G contains a copy of F 2k−1 .
From now on we let F be an arbitrary fixed multigraph and write H = F 2k−1 . Moreover, throughout the paper we tacitly assume that n is sufficiently large.
The next definition was introduced in [3], and was used to prove Theorem 1.4. The next lemma will be used several times later.
Lemma 2.5. Let ℓ ≥ 2 and let L > ℓ. If a path P = v 0 . . . v ℓ is L-admissible, but not L-good, then there exist at least L pairwise internally vertex-disjoint paths of length ℓ from v 0 to v ℓ .
Proof. Take a maximal set of pairwise internally vertex-disjoint paths of length ℓ from v 0 to v ℓ and assume that it consists of fewer than L paths. These paths contain at most L(ℓ − 1) internal vertices in total and any path of length ℓ between v 0 and v ℓ intersects at least one of these vertices. Since there are at least L 5 ℓ L-admissible paths of length ℓ between v 0 and v ℓ , it follows by pigeon hole that there exist some 1 ≤ i ≤ ℓ − 1 and some x ∈ V (G) such that there are at least so either there are more than L 5 i L-good paths of length i between v 0 and x or there are more than L 5 ℓ−i L-good paths of length ℓ − i between x and v ℓ . In either case, we contradict the definition of an L-good path.
Our strategy will be to prove that, roughly speaking, in any almost regular H-free graph there are many good paths of length 2k. As we will see in Section 3, the techniques in [3] can be easily applied to prove this for paths of length k. The novelty of this paper is the machinery that allows us to extend this to longer paths, using very different techniques. This is given in Section 4, where we prove the following lemma. Lemma 2.6. Let G be an H-free K-almost-regular graph on n vertices with minimum degree δ ≥ L 100 k |V (H)| , and let S ⊂ V (G). Then, provided that L is sufficiently large compared to |V (H)| and K, |S| = ω( n δ 1/2 ) and |S| = ω( n L 1/2 ), the number of L-good paths of length 2k with both endpoints in S is Ω( |S| 2 δ 2k n ).
Note that in this result and everywhere else in the paper, the asymptotic notation Ω allows the implied constant to depend on k, |V (H)| and K, which are thought of as constants, while δ and L are functions of n.
With Lemma 2.6 in hand, the proof of Theorem 2.2 is immediate.
. Then we may apply Lemma 2.6 with S = V (G) to get that the number of L-good paths of length 2k in G is Ω(nδ 2k ), which is ω(n 2 f (2k, L)). However, by the definition of Lgoodness, between any two vertices there can be at most f (2k, L) such paths, which is a contradiction.
The proof of Theorem 2.3 is slightly more complicated, and it uses ideas from [8].
Proof of Theorem 2.3. Firstly note that F is a subgraph of K t for some t, so it suffices to prove the result for F = K t . Let ε > 0 be sufficiently small, to be specified, and let G be a K-almost-regular graph on n vertices with minimum degree δ = ω(n 1 2k −ε ). Assume that G does not contain a copy of H = F 2k−1 .
For vertices u, v ∈ V (G), let us write u ∼ v if there is a path of length 2k between u and v. Also, let us say that u and v are distant if for every 1 ≤ i ≤ 4k − 2, the number of walks of length i between u and v is at most δ i−2k+1/2 . Observe that for any u ∈ V (G) the number of walks of length i starting from u is at most (Kδ) i , so the number of vertices v ∈ V (G) for which there are at least δ i−2k+1/2 walks of length i from u to v is at most Define c 0 = ε and c ℓ+1 = (3 · 5 2k + 1)c ℓ + 2kε for 0 ≤ ℓ ≤ t − 1. Assume that ε is small enough so that for all 0 ≤ ℓ ≤ t. Then in particular c ℓ ≤ 1 4k − ε/2 holds for all 0 ≤ ℓ ≤ t. For future reference, note that then (iv) x i and x j are distant for every 1 ≤ i < j ≤ ℓ.
Note that in particular for ℓ = t, condition (i) guarantees the existence of a subgraph K 2k−1 t , so it suffices to prove the claim.
Proof of Claim. We proceed by induction on ℓ. For ℓ = 0, we may take S 0 = V (G). Assume now that we have verified the claim for ℓ.
Suppose that for some y ∈ S ℓ there exist 1 ≤ i < j ≤ ℓ and two paths of length 2k, one (called P i ) from x i to y and one (called P j ) from x j to y, which share a vertex other than y. Let they intersect at some vertex z = y. Now let the subpath of P i between x i and z have length α and let the subpath of P j between x j and z have length β. Then there is a walk of length α + β from x i to x j through z. Moreover, there is a path of length 2k − α from z to Let Y be the set of y ∈ S ℓ for which there exist some 1 ≤ i < j ≤ ℓ and a walk W of length γ ≤ 4k − 2 between x i and x j such that for some vertex w on W the distance of y from w is at most 4k − γ − 1. By condition (iv), there are at most δ γ−2k+1/2 walks of length γ between any x i and x j so there are O(δ γ−2k+1/2 ) vertices appearing in at least one of these walks. Therefore the number of vertices at distance at most 4k − γ − 1 from at least one of these vertices is Notice that by the discussion above, for any y ∈ S ℓ \ Y and any i = j, a path of length 2k from x i to y, and a path of length 2k from x j to y have no common vertex other than y. Thus, by condition (ii) there exist ℓ paths of length 2k, one from each x i to y which are pairwise vertex-disjoint apart from at y. Moreover, these paths are also vertex-disjoint from the paths forming the K 2k−1 ℓ guaranteed by condition (i), apart from the trivial intersections at x 1 , . . . , x ℓ (else, there is a path of length at most 2k − 1 from y to a point on a path of length 2k between some x i and x j , which contradicts the fact that y ∈ Y ). Thus, for any y ∈ S ℓ \ Y there is a copy of K 2k−1 ℓ+1 in G with the vertices of the subdivided K ℓ+1 being x 1 , . . . , x ℓ , y.
Let Z be the set of z ∈ S ℓ which are not distant to x i for at least one 1 ≤ i ≤ ℓ. By the second paragraph in this proof,  (2), we have n 1−c ℓ = ω( n δ 1/2 ), and by the definition of L, we have n 1−c ℓ = ω( n L 1/2 ). Hence, by Lemma 2.6, the number of L-good paths of length 2k with both endpoints in S ′ ℓ is Ω( . Between any two vertices in S ′ ℓ there are at most f (2k, L) L-good paths of length 2k, so the number of pairs (z, y) ∈ S ′ ℓ × S ′ ℓ with z ∼ y is Ω( nf (2k,L) ). Thus, there exists some x ℓ+1 ∈ S ′ ℓ such that the number of y ∈ S ′ ℓ with x ℓ+1 ∼ y is Ω( . Set S ℓ+1 to be the set of these y ∈ S ′ ℓ , and note that properties (i)-(iv) are satisfied for ℓ + 1.

Short paths
Our aim in this section is to prove the following lemma.
Lemma 3.1. Let G be an H-free K-almost-regular graph on n vertices with minimum degree δ ≥ L 100 k |V (H)| . Then, provided that L is sufficiently large compared to |V (H)| and K, the number of paths of length k that are not good is O( nδ k L ). The proof of this is almost identical to that of Lemma 6.4 in [3], nevertheless we include it here for completeness and since some minor details need to be modified.
The next definition is for notational convenience. In what follows, for v ∈ V (G), we shall write Γ i (v) for the set of vertices u ∈ V (G) for which there exists a path of length i from v to u and write N (v) = Γ 1 (v). The next lemma is a slight variant of Lemma 6.7 from [3]. Lemma 3.3. Let 2 ≤ ℓ ≤ k and 1 ≤ i ≤ ℓ. Let G be a K-almost-regular graph on n vertices with minimum degree δ > 0. Let X, Y, Z ⊂ V (G) be such that |Z| ≤ L 1/10 , |Y | ≤ (Kδ) ℓ−1 and, for any x ∈ X, the number of y ∈ Y such that (x, y) is (ℓ, L)-bad is as at least (Kδ) ℓ−1 f (ℓ−1,L) 2 . Then, provided that L is sufficiently large compared to k and K, there exist a path of length 2i in G, disjoint from Z, whose endpoints form a set R ⊂ Y , and a subset X ′ ⊂ X such that |X ′ | ≥ |X \ Z|/(16f (ℓ − 1, L) 2 ) and (x ′ , r) is (ℓ, L)-bad for every x ′ ∈ X ′ and r ∈ R.
The number of such paths intersecting Z is at most |Z|ℓ(Kδ) ℓ−1 . Indeed, there are at most |Z| choices for the element of Z in the path, at most ℓ choices for its position in the path and, given a fixed choice for these, at most (Kδ) ℓ−1 choices for the other ℓ − 1 vertices in the path. (Note that as X∩Z = ∅, the vertex in Z is not x * .) But |Z|ℓ(Kδ) ℓ−1 ≤ L 1/10 ℓK ℓ−1 δ ℓ−1 , so, for L sufficiently large there are Ω(f (ℓ − 1, L) 3 δ ℓ−1 ) paths of length ℓ starting at x * and ending in Y ′ that avoid Z. Moreover, since |Γ ℓ−i (x * )| ≤ (Kδ) ℓ−i , it follows that there exists some u ∈ Γ ℓ−i (x * ) such that there are Ω(f (ℓ − 1, L) 3 δ i−1 ) paths of length i from u to Y ′ , all avoiding Z.
Take now a maximal set of such paths which are pairwise vertex-disjoint apart from at u. We claim that there are Ω(f (ℓ − 1, L) 3 ) such paths. Suppose otherwise. Then all the Ω(f (ℓ − 1, L) 3 δ i−1 ) paths of length i from u to Y ′ intersect a certain set of size o(f (ℓ − 1, L) 3 ) not containing u. But there are o(f (ℓ − 1, L) 3 )δ i−1 such paths, which is a contradiction.
So we have r = Ω(f (ℓ−1, L) 3 ) paths P 1 , . . . , P r of length i from u to Y ′ which are pairwise vertex-disjoint except at u and avoid Z. Let the endpoints of these paths be y 1 , . . . , y r . Since y j ∈ Y ′ for all j, the number of pairs (x, y j ) with x ∈ X which are (ℓ, L)-bad is at least r|X| 2f (ℓ−1,L) 2 . Therefore, by Jensen's inequality, for an average x ∈ X there are at least r/(2f (ℓ−1,L) 2 ) 2 choices 1 ≤ j 1 < j 2 ≤ r such that both (x, y j 1 ) and (x, y j 2 ) are (ℓ, L)-bad.
and (x, y j 2 ) are (ℓ, L)-bad} has size at least |X|/(4f (ℓ − 1, L) 2 ) 2 . We can now take R = {y j 1 , y j 2 }, and the union of the paths P j 1 and P j 2 is a suitable path of length 2i.
The following lemma is a small modification of Lemma 6.8 from [3]. Proof. Suppose otherwise. Let Y = Γ ℓ−1 (v) and note that |Y | ≤ (Kδ) ℓ−1 . For any x ∈ N (v) and any y ∈ Y , the number of L-admissible paths of the form xvv 2 . . . v ℓ−1 y is at most f (ℓ − 1, L). Indeed, in any such path, the subpath vv 2 v 3 . . . v ℓ−1 y is L-good, and for any fixed y ∈ Y there are at most f (ℓ − 1, L) such L-good paths. Hence, by assumption, the number of pairs (x, y) ∈ N (v) × Y such that there is an L-admissible, but not L-good, path of the form xvv 1 . .
Our aim now is to find a copy of H in G, which will yield a contradiction. Write k = jℓ+i with 1 ≤ i ≤ ℓ.
Note that if L is sufficiently large, then so we may apply Lemma 3.3 repeatedly |E(F )| + |V (H)| ≤ 2|V (H)| times and still get a set X ′ of size at least L. Thus, we find disjoint paths P e of length 2i for every e ∈ E(F ) whose endpoint sets are R e ⊂ Y , and sets X final ⊂ X and U ⊂ Y with |X final | = |U | = |V (H)| such that V (P e ), X final and U are pairwise disjoint and any pair (x, y) with x ∈ X final and y ∈ U ∪ e∈E(F ) R e is (ℓ, L)-bad. For e ∈ E(F ), let y e −k y e −k+1 . . . y e k be the path of length 2k replacing the edge e.
A copy of H in G can now be constructed as follows. For each e ∈ E(F ), map the path y e −i y e −i+1 . . . y e i to P e . Then map, for each e ∈ E(F ), the vertices y e i+ℓ , y e −(i+ℓ) to X final in an arbitrary injective manner. Also, map each y e i+2ℓ , y e −(i+2ℓ) to U in an arbitrary injective manner. More generally, map the vertices y e i+aℓ , y e −(i+aℓ) with a ≥ 1 odd to X final in an arbitrary injective manner and map the vertices y e i+aℓ , y e −(i+aℓ) with a ≥ 2 even to U in an arbitrary injective manner. We then just need to find paths of length ℓ connecting y e i+aℓ and y e i+(a+1)ℓ (and paths of length ℓ connecting y e −(i+aℓ) and y e −(i+(a+1)ℓ) ) which are disjoint from each other and from the images of the already mapped vertices. Since (x, y) is (ℓ, L)-bad for every x ∈ X final and y ∈ U ∪ e∈E(F ) R e , such paths exist by Lemma 2.5, provided that L is sufficiently large.
Corollary 3.5. Let G be an H-free K-almost-regular graph on n vertices with minimum degree δ ≥ L 100 k |V (H)| . Then, provided that L is sufficiently large compared to |V (H)| and K, for any 2 ≤ ℓ ≤ k, the number of L-admissible, but not L-good, paths of length ℓ is at most n 2(Kδ) ℓ f (ℓ−1,L) . Now we are in a position to prove Lemma 3.1.
Proof of Lemma 3.1. Suppose that the path u 0 u 1 . . . u k is not L-good. Take 0 ≤ i < j ≤ k with j − i minimal such that u i u i+1 . . . u j is not L-good. Then u i . . . u j is Ladmissible. For any fixed i, j, by Corollary 3.5, the number of such paths is at most Using that i and j can take at most k + 1 values each, it follows that the number of not L-good paths of length k is at most (k + 1) 2 4K k · nδ k L .

Long paths
In what follows, for a vertex x ∈ V (G) and a nonnegative integer i, we write P i (x) for the set of directed paths of length i starting at x. For an element P ∈ P i (x), we let v(P ) be the endpoint of the path P .
Lemma 4.2. Let G be a graph with maximum degree at most Kδ. Let x, y ∈ V (G) and let i, j be nonnegative integers with i + j < 2k. If (x, y) is (i, j)-rich, then there exist |V (H)| pairwise internally vertex-disjoint paths of length 2k between x and y.
Proof. Choose a maximal set of pairwise internally vertex-disjoint paths R 1 , . . . , R α between x and y and assume that α < |V (H)|. Let T be the set of the vertices appearing in at least one of these paths. Note that |T | < |V (H)|(2k + 1).
Claim. If there is a pair (P, Q) ∈ P i (x) × P j (y) such that A path provided by the claim would contradict the maximality of R 1 , . . . , R α , so it suffices to prove that there are paths P, Q satisfying (i)-(iv) above.
Since the maximum degree of G is at most Kδ, the number of paths of length i − 1 in G intersecting T is at most i|T |(Kδ) i−1 , so the number of P ∈ P i (x) which have a vertex in T \ {x} is at most 2i|T |(Kδ) i−1 . Since |P j (y)| ≤ (Kδ) j , the number of pairs (P, Q) ∈ P i (x) × P j (y) failing condition (i) above is at most 2i|T |(Kδ) i−1 (Kδ) j . Similarly, the number of pairs failing (ii) is at most 2j|T |(Kδ) j−1 (Kδ) i . Finally, for every P ∈ P i (x), the number of paths of length j − 1 which intersect P is at most (i + 1)j(Kδ) j−1 , so the number of pairs (P, Q) ∈ P i (x) × P j (y) for which P and Q share a vertex other than y is at most (Kδ) i · 2(i + 1)j(Kδ) j−1 . So the number of pairs which fail at least one of (i),(ii),(iii) is at most (2(i + j)|T | + 2(i + 1)j)(Kδ) i+j−1 ≤ (2(i + j)|V (H)|(2k + 1) + 2(i + 1)j)(Kδ) i+j−1 . By the definition of (i, j)-richness of (x, y) it follows that there is a pair (P, Q) satisfying (i)-(iv). Definition 4.3. For a vertex v ∈ V (G) and some 1 ≤ ℓ ≤ k, define an auxiliary graph G ℓ (v) as follows. The vertices of G ℓ (v) are the (k + 1)-tuples (u 0 , u 1 , . . . , u k ) ∈ V (G) k+1 with u 0 = v such that u i u i+1 ∈ E(G) for all i. Vertices (u 0 , . . . , u k ) and (u ′ 0 , . . . , u ′ k ) are joined by an edge if v, u 1 , u 2 , . . . , u k , u ′ 1 , . . . , u ′ k are distinct and there exist 0 ≤ i, j ≤ k − 1 such that the pair (u ℓ , u ′ ℓ ) is (i, j)-rich. Since the vertex set of G ℓ (v) does not depend on ℓ, we may define G(v) to be the union 1≤ℓ≤k G ℓ (v). Lemma 4.4. Let G be a graph with maximum degree at most Kδ which does not contain H as a subgraph. Let t = |V (F )|. Then for any v ∈ V (G) and any 1 ≤ ℓ ≤ k, the graph G ℓ (v) is K t -free.
Moreover, let r = R k (t) be the k-colour Ramsey number. Then G(v) is K r -free.
Proof. Suppose that G ℓ (v) contains K t as a subgraph. Let the corresponding vertices be the vectors u 1 , . . . , u t . Let their respective (ℓ + 1)th coordinate be u 1 ℓ , . . . , u t ℓ . For every a = b, since u a u b is an edge in G ℓ (v), it follows that u a ℓ and u b ℓ are distinct, and, by Lemma 4.2, there exist |V (H)| pairwise internally vertex-disjoint paths of length 2k between them. It is not hard to see that this implies that there is a copy of H in G in which the vertices of F are mapped to u 1 ℓ , . . . , u t ℓ . This is a contradiction, so G ℓ (v) is indeed K t -free. Suppose there is a copy of K r in G(v). Then each edge in this K r can be coloured with one of the colours 1, 2, . . . , k such that if an edge gets colour i, then it lies in G i (v). By the definition of r, there exists a monochromatic K t in this k-edge-coloured K r , which gives a K t in some G ℓ (v), contradicting the first paragraph.
The next lemma provides us a large set of walks of length 2k with both endpoints in S. Later, we will argue that most of them are L-good paths.
Lemma 4.5. Let r = R k (t) denote the k-colour Ramsey number where t = |V (F )|. Let G be an H-free K-almost-regular graph on n vertices with minimum degree δ and let S ⊂ V (G) such that |S| ≥ 2nr/δ k . Then there are at least |S| 2 δ 2k 4r 2 n vectors (u −k , . . . , u k ) ∈ V (G) 2k+1 with the following properties Proof. Since the minimum degree of G is δ, the number of (k + 1)-tuples (v 0 , v 1 , . . . , v k ) ∈ V (G) k+1 with v k ∈ S and v i v i+1 ∈ E(G) for every 0 ≤ i ≤ k − 1 is at least |S|δ k . Writing T (v 0 ) for the set of such vectors for a fixed v 0 and letting g(v 0 ) = |T (v 0 )|, we get that Note that T (v 0 ) ⊂ V (G(v 0 )). By Lemma 4.4, the graph G(v 0 )[T (v 0 )] is K r -free. This graph has g(v 0 ) vertices, so if g(v 0 ) ≥ r, then the number of non-edges in G(v 0 )[T (v 0 )] is at least are such that vv ′ is not an edge in G(v 0 ), then (u −k , . . . , u k ) = (v ′ k , v ′ k−1 , . . . , v ′ 1 , v 0 , v 1 , . . . , v k ) satisfies all three properties in the statement of the lemma. Therefore the number of such (2k + 1)-tuples with u 0 = v 0 is at least g(v 0 ) 2 r 2 provided that g(v 0 ) ≥ r. By (3) and Jensen's inequality, we get v 0 ∈V (G):g(v 0 )≥r g(v 0 ) 2 r 2 ≥ |S| 2 δ 2k 4r 2 n , and the proof is complete. The following simple lemma shows that most walks of length 2k are paths.
Lemma 4.6. Let G be a graph on n vertices with maximum degree at most Kδ. Then the number of (2k + 1)-tuples (u −k , . . . , u k ) ∈ V (G) 2k+1 such that u i u i+1 ∈ E(G) for every i and u i = u j for some i = j is at most 2k+1 2 K 2k−1 · nδ 2k−1 .
of possible ways to extend to u −α u −α+1 . . . u β , and the second factor bounds the number of possible ways to extend that to u −k u −k+1 . . . u k . The number of possible choices for u −ℓ u −ℓ+1 . . . u ℓ is O(nδ 2ℓ ), so the result follows.
We are now in a position to complete the proof of Lemma 2.6.