Ramsey numbers upon vertex deletion

Given a graph G $G$ , its Ramsey number r( G ) $r(G)$ is the minimum N $N$ so that every two‐coloring of E( K N ) $E({K}_{N})$ contains a monochromatic copy of G $G$ . It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from G $G$ , the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super‐constant factor. One consequence of this result is the following. There exists a family of graphs { G n } $\{{G}_{n}\}$ so that in any Ramsey coloring for G n ${G}_{n}$ (i.e., a coloring of a clique on r( G n ) − 1 $r({G}_{n})-1$ vertices with no monochromatic copy of G n ${G}_{n}$ ), one of the color classes has density o( 1 ) $o(1)$ .


Introduction
The Ramsey number of a graph G is the minimum N so that every two-coloring of the edges of the complete graph K N contains a monochromatic copy of G. Motivated by understanding the concentration of the Ramsey number of the Erdős-Rényi random graph G(n, p), Conlon, Fox, and Sudakov [3] made the following natural conjecture.

Conjecture 1.1 ([3, Conjecture 5.1]
).There exists a universal constant C > 0 so that the following holds.Let G be a graph, and let H be obtained from G by deleting a single vertex.Then r(G) ≤ C • r(H).
Conlon, Fox, and Sudakov proved two natural weakenings of this conjecture.First, they showed that Conjecture 1.1 holds when G is a dense graph, i.e. when it has n vertices and at least p n 2 edges, and C is allowed to depend on p.They also proved a slightly weaker inequality, namely that for any (n + 1)-vertex graph G and any graph H obtained by deleting a single vertex from G, one has r(G) ≤ 2n • r(H).
Our goal in this paper is to disprove Conjecture 1.1.We find an explicit family of graphs such that deleting a single vertex decreases the Ramsey number by a super-constant factor.Here, and in the rest of the paper, all logarithms have base 2.
Theorem 1.2.For any n ≥ 16, there exists an (n + 1)-vertex graph G with Ramsey number r(G) > 1  3 n log n.However, there is a vertex of G whose deletion yields a graph H with Ramsey number r(H) = n.In particular, r(G) = ω(r(H)).
In the other direction, we prove the following strengthening of (1).
Theorem 1.3.Let G be an (n+1)-vertex graph, and suppose that H is obtained from G by deleting one vertex.Then r(G) ≤ C n log n • r(H) for some absolute constant C > 0.
Recall that the edge density of an N -vertex graph is its number of edges divided by N 2 .For ε ∈ (0, 1  2 ], let us say that a two-coloring of E(K N ) is ε-balanced if both color classes have edge density at least ε.Ramsey properties of ε-balanced colorings have been well studied, see e.g.[1,4,6,7,9].A Ramsey coloring for a graph G is a two-coloring of E(K N ), where N = r(G) − 1, which contains no monochromatic copy of G.In general, there may be multiple non-isomorphic Ramsey colorings for G.For ε ∈ (0, 1  2 ], let us say that G is ε-Ramsey-balanced if there exists a Ramsey coloring for G which is ε-balanced.
Our next result shows that for Ramsey-balanced graphs G, Conjecture 1.1 is true.
Proposition 1.4.Let ε ∈ (0, 1  2 ], let G be an ε-Ramsey-balanced graph, and let H be obtained from G by deleting one vertex.Then Proposition 1.4 shows that Conjecture 1.1 is true for any graph G which is Ω(1)-Ramseybalanced.Combined with Theorem 1.2, we conclude that for any fixed ε > 0, there exist graphs which are not ε-Ramsey-balanced.This is somewhat surprising, as one might naively expect all graphs to be Ramsey-balanced: since we are searching for the same graph in both the red and blue colors, it seems plausible that any extremal coloring should have roughly equal red and blue densities, or at least that these densities would be within a constant factor of one another.
The rest of this paper is organized as follows.In Section 2, we exhibit explicit graphs in which the removal of one vertex changes the Ramsey number by a super-constant factor, proving Theorem 1.2.In Section 3, we prove Theorem 1.3, establishing a stronger upper bound than (1).In Section 4, we prove Proposition 1.4, showing that Conjecture 1.1 is true for Ramsey-balanced graphs.In Section 5, we make some comments on the multicolor version of these problems.We end with some concluding remarks and open problems in Section 6.For clarity of presentation, we systematically omit floor and ceiling signs whenever they are not crucial.

Proof of Theorem 1.2
Let k and n ≥ 4 k be positive integers.Let H k,n be the n-vertex graph consisting of the clique K k , plus n − k isolated vertices.Additionally, let G k,n be obtained from H k,n by adding an apex vertex, i.e. a vertex adjacent to all vertices of H k,n .We note for future reference that G k,n is connected and has chromatic number k + 1. Theorem 1.2 follows from the following two lemmas, both of which have simple proofs using standard techniques, which we give after stating the lemmas.
We remark that the graph 2) is tight up to the constant factor for the graphs G k,n and H k,n .This implies that if one were to search for stronger counterexamples to Conjecture 1.1, they would need to have somewhat higher degeneracy (i.e.d = ω(log n)).
Proof of Theorem 1.3.Note that for d ≥ 1, we have Setting C = 2 √ A and applying Proposition 3.1 yields Theorem 1.3.
To prove Proposition 3.1, we need the following simple and well-known lemma, which shows that one can embed a graph of bounded degeneracy in a very dense graph.Lemma 3.2.Let H be an n-vertex d-degenerate graph, and let R be an N -vertex graph in which every vertex has at most (N − n)/d non-neighbors.Then H is a subgraph of R.
Proof.Let the vertices of H be v 1 , . . ., v n , sorted so that each v i has at most d neighbors v j with j < i.We inductively pick an embedding ϕ : V (H) → V (R), maintaining the property that ϕ(v 1 ), . . ., ϕ(v i ) span a copy of H[v 1 , . . ., v i ].First, we let ϕ(v 1 ) be an arbitrary vertex of R. Suppose we've defined ϕ(v 1 ), . . ., ϕ(v i−1 ), and we wish to define ϕ(v i ).Let J be the set of j < i with v j v i ∈ E(H), so that |J| ≤ d.For every j ∈ J, there are at most (N − n)/d non-neighbors of ϕ(v j ) in R, and thus at most N − n vertices of R are non-adjacent to ϕ(v j ) for some j ∈ J. Thus, there are at least n vertices of R that are adjacent to all {ϕ(v j )} j∈J , and fewer than n of these vertices have been used in embedding v 1 , . . ., v i−1 .So we pick any of the remaining candidate vertices as ϕ(v i ), which maintains our inductive hypothesis.Continuing in this manner until i = n, we find a copy of H in R.
We will use the following result of Erdős and Szemerédi [6].It shows that colorings which are not ε-balanced contain larger cliques than what is implied by the Ramsey number alone.

Lemma 3.3 ([6, Theorem 2]
).There exists an absolute constant a > 0 such that the following holds for all ε ∈ (0, 1  2 ] and every positive integer N .Any two-coloring of E(K N ) which is not ε-balanced contains a monochromatic clique of order a ε log 1 ε log N .
Finally, we will need the following well-known lower bound on the Ramsey number of a graph of high degeneracy (see e.g.[8, Section 11, Remark 2]).We provide the proof for completeness.
Proof.By the definition of degeneracy, there exists a subgraph G 0 ⊆ G with minimum degree at least d.Let G 0 have k vertices, so that it has at least dk/2 edges, and note that k ≥ 2 since d ≥ 1.Let N = 2 d/2 , and consider a uniformly random two-coloring of E(K N ).The expected number of monochromatic copies of G 0 is at most Thus, there exists a two-coloring of E(K N ) with fewer than 2 monochromatic copies of G 0 .By deleting one vertex, we obtain a coloring on N − 1 vertices with no monochromatic copies of G 0 , and thus, no monochromatic copies of G.This shows that r(G) ≥ (N − 1) + 1 = N , as claimed.
We are now ready to prove Proposition 3.1.
Proof of Proposition 3.1.Let N = r(G) − 1, and fix a two-coloring of E(K N ) with no monochromatic copy of G. Let v be a vertex of K N .Assume without loss of generality that v has at least (N − 1)/2 red neighbors, and let S be this red neighborhood.Now consider the induced coloring on S. Let w be a vertex of maximum blue degree in this induced coloring, and let T be the blue neighborhood of w in S. Then every vertex in T is adjacent to v in red and to w in blue, and thus T cannot contain a monochromatic copy of H. Therefore, |T | ≤ r(H) − 1.We now lower-bound |T | in two different ways.First, we claim that |T | > (|S| − n)/d.Since w is a vertex of maximum blue degree, we see that if this is not the case, then every vertex in S has fewer than (|S| − n)/d blue neighbors in S.
Applying Lemma 3.2 with R being the red graph on S, we find that there is a red copy of H in S. Together with v, this yields a red copy of G, a contradiction.This proves the claim, which implies that , by our assumption that d ≥ 10.Note too that by choosing c sufficiently small, we can ensure that This shows that S contains a monochromatic clique of order 2n, and thus a copy of G, a contradiction.This proves the claim that |T | ≥ ε(|S| − 1), which implies that for a constant A = 4/c > 0.

Ramsey-balanced graphs
In this section we discuss Ramsey-balanced graphs and prove Proposition 1.4, which says that Conjecture 1.1 is true for Ramsey-balanced graphs.As discussed in the Introduction, this shows that there exists a sequence of graphs G n such that in any Ramsey coloring for G n , one of the color classes has density o(1).
For a vertex v in a two-colored complete graph, we denote by N R (v), N B (v) the red and blue neighborhoods of v, respectively.The only property we need of ε-balanced colorings is that in any ε-balanced coloring, we can find vertices v, w such that N R (v) ∩ N B (w) is large.Such a statement can be easily deduced from [7, Lemma 2.1]; however, the following result, due to Kamčev and Müyesser1 , gives a stronger quantitative bound.Lemma 4.1 (Kamčev-Müyesser).Let 0 < ε ≤ 1 2 , and fix an ε-balanced coloring of E(K N ) with colors red and blue.Then there exist v, w ∈ In the course of the proof of Lemma 4.1, we will need the following simple fact about convex functions, which can be viewed as a "reversed" form of Jensen's inequality.Although this result appears to be well-known, we were not able to find a concise statement in the literature, and include the short proof for a completeness.
Proof.The definition of convexity implies that for every 1 ≤ i ≤ N .Summing this up over all i gives the claimed result.
We are now ready to prove Lemma 4.1.
Proof of Lemma 4.1.By swapping the roles of the colors if necessary, we may assume without loss of generality that at least half the edges are colored blue.We may also assume that exactly ε N 2 edges are red, and the remainder are blue; if this is not the case, then we simply replace ε by a larger value, and obtain a stronger conclusion.Thus, the average red degree is ε(N − 1), and the average blue degree is (1 − ε)(N − 1).Suppose first that there is some vertex v with |N R (v)| ≥ 3 2 ε(N − 1).Let w be a vertex of maximum blue degree, so that |N B (w)| ≥ (1 − ε)(N − 1).Then by the pigeonhole principle, we have that Therefore we may assume that every vertex of K N has red degree at most 3 2 ε(N − 1).Applying Lemma 4.2 with f (y) = y 2 and s = 3  2 ε(N − 1), this implies that where the final inequality uses that ε ≤ 1 2 .Let X denote the number of copies of K 1,2 with one red and one blue edge.We have that On the other hand, we have that where the second equality is because With these preliminaries, Proposition 1.4 is an easy consequence.
Proof of Proposition 1.4.Let N = r(G) − 1, and fix an ε-balanced coloring of E(K N ) with no monochromatic copy of G. Lemma 4.1 yields two vertices v, w . We claim that S contains no monochromatic copy of H. Indeed, if there were some red copy of H in S, then by adding v to it we would find a red copy of G; similarly, a blue copy of H in S yields a blue copy of G by adding w.So we conclude that |S| ≤ r(H) − 1, which implies that To conclude this section, we remark that the results of Section 3 can be more or less equivalently phrased in terms of Ramsey-balanced graphs.Namely, the proof of Proposition 3.1 can be used to show that if H is an n-vertex graph of degeneracy d ≥ 1, then H is both Ω( 1 d )-Ramseybalanced and Ω(d/(n log n d ))-Ramsey-balanced.Balancing these two bounds and plugging them into Proposition 1.4 yields Theorem 1.3.

More colors
For an integer q ≥ 2, let r(H; q) denote the q-color Ramsey number of H, that is, the least N so that every q-coloring of E(K N ) contains a monochromatic copy of H.When the number of colors is greater than 2, we can prove a stronger version of Theorem 1.2, showing a polynomial gap between r(G) and r(H).
Theorem 5.1.Fix an integer q ≥ 3.For all sufficiently large n, there exists an (n + 1)-vertex graph G with Ramsey number r(G; q) > n (1) .However, there is a vertex of G whose deletion yields a graph H with Ramsey number r(H; q) = n.In particular, for sufficiently large n, r(G; q) ≥ (r(H; q)) 1+α for some fixed α > 0 depending only on q.
Proof.Fix positive integers k and n ≥ q qk , and let H k,n and G k,n be the graphs defined in Section 2. As in the proof of Lemma 2.1, it is easy to check that r(H k,n ; q) = n, by using the well-known upper bound r(K k ; q) ≤ q qk .
For the lower bound on r(G k,n ; q), let m = r(K k+1 ; q − 1) − 1 and N = mn.By the definition of r(K k+1 ; q − 1), there exists a coloring χ : E(K m ) → [q − 1] with no monochromatic copy of K k+1 .We partition V (K N ) into m blocks S 1 , . . ., S m , each comprising n vertices.For all i = j, we color all edges between S i and S j by the color χ(i, j) ∈ [q − 1], and we color all edges inside some part with color q.There is no copy of K k+1 in any of the first q − 1 colors, and thus no copy of G k,n in any of these colors.Additionally, G k,n is connected and has n + 1 vertices, so there is no copy of G k,n in the qth color either.This shows that r(G k,n ; q) > N .
However, as the value of α we obtain is likely far from optimal in any case, we chose to use the somewhat simpler expression from [12].One strange feature of Theorem 5.1 is that the bound on α actually deteriorates as q gets larger, since the expression (3q −5)/(8q log q) tends to 0 as q → ∞.However, this is really a consequence of our poor understanding of multicolor Ramsey numbers.Indeed, the best known bounds on r(K k ; q) are cqk ≤ log r(K k ; q) ≤ qk log q for some constant c > 0. The logarithmic gap between the lower and upper bounds here appears as the factor of log q in the denominator of α.As such, we expect that a better understanding of the asymptotics of r(K k ; q) would lead to a bound on α which does not deteriorate as q → ∞.
While the lower bound of Theorem 5.1 is noticeably stronger than that of Theorem 1.2-we obtain a polynomial rather than barely super-linear bound when q ≥ 3-unfortunately, the upper bounds in the multicolor case are much worse.Indeed, mimicking the proof 2 of (1), one obtains that if G is an n-vertex graph and H is obtained from G by deleting a single vertex, then r(G; q) ≤ 2 cqn r(H; q) for an absolute constant c q > 0 depending only on q, for any fixed q ≥ 3.In other words, in contrast to the O(n) bound on r(G)/r(H) in (1), in the case of q ≥ 3, the bound on r(G; q)/r(H; q) is exponential in n.Given that r(G; q) is itself at most exponential in n, this bound is extremely weak, and it is not clear how to meaningfully improve it.

Concluding remarks
Although Theorem 1.2 disproves the original conjecture of Conlon, Fox, and Sudakov, there remain a number of other interesting open problems.
The most natural question is to close the gap between Theorems 1.2 and 1.3.Formally, we can define : G is an (n + 1)-vertex graph and v ∈ V (G) , which measures how much the deletion of a single vertex can affect the Ramsey number of an (n + 1)-vertex graph.Then Theorems 1.2 and 1.3 imply that Ω(log It would be interesting to improve either bound; concretely, we make the following conjecture. Conjecture 6.1.There is some c > 0 so that f (n) ≥ n c for all sufficiently large n.
It would also be very interesting to improve the corresponding gap for multicolor Ramsey numbers; as discussed in Section 5, the bounds for q ≥ 3 are very far apart.Additionally, the structure of the counterexamples to Conjecture 1.1 is somewhat dissatisfying, since H k,n consists of a small clique and an enormous number of isolated vertices.One can avoid using isolated vertices (for example, essentially the same proof 3 goes through if one lets H ′ k,n be the disjoint union of n/k copies of K k ), but there is a limit to how far one can push this using the techniques of this paper.Namely, we do not know how to prove a result like Theorem 1.2 unless the graph H consists of a large number of small connected components.In particular, it seems that the following conjecture would require new ideas.Conjecture 6.2.There exists a graph G and a vertex v ∈ V (G) such that H = G\{v} is connected, and r(G) = ω(r(H)).
It would also be interesting to study an "average-case" version of this question, rather than the "worst-case" version considered in Conjecture 1.1 and Theorems 1.2 and 1.3.For a graph G, 2 The key observation is that, in a q-coloring of KN , we may assume that the density of every color is at least 1/r(G; q − 1), as otherwise we may apply Turán's theorem to find a set of r(G; q − 1) vertices colored by only q − 1 colors, which then contains a monochromatic copy of G. Applying this observation q times in nested common neighborhoods, we can find a set of N/r(G; q − 1) q vertices, which is complete in each of the q colors to some other vertex.Hence r(G; q)/r(G; q − 1) q < r(H; q), which implies the claimed bound since r(G; q − 1) q ≤ 2 cq n . 3The only material difference is that now, one needs to replace Lemma 2.1 by the weaker bound r(H ′ n,k ) ≤ 3n (assuming n ≥ 4 k ).This bound is proved by repeatedly applying the Erdős-Szekeres bound to find disjoint monochromatic K k , continuing as long as at least n vertices remain.One can thus find 2n/k disjoint monochromatic K k , half of which must have the same color, yielding a monochromatic H ′ n,k .

1 ε
and thus N < 2dr(H) − 2d + 2n + 1 ≤ 2dr(H) + 2n − 1 ≤ 4dr(H) since d ≥ 1 and since n ≤ r(H).Recalling that N = r(G) − 1 yields (2).Note that (3) follows from (1) if d ≤ 9 by choosing A sufficiently large, so we henceforth assume d ≥ 10.We now claim that |T | ≥ ε(|S| − 1), where ε = cd/(n log n d ), for some sufficiently small constant c > 0. If this is not the case, then every vertex in S has blue degree less than ε(|S| − 1), and thus the blue edge density in S is less than ε.So the induced coloring on S is not ε-balanced, and by Lemma 3.3, we may find in S a monochromatic clique of order a ε log log |S|.Note that, as H ⊆ G, the degeneracy of G is at least d, and therefore Lemma 3.4 implies that
[5]of of Lemma 2.1.The lower bound r(H k,n ) ≥ n is immediate since every n-vertex graph has Ramsey number at least n.For the upper bound, fix a two-coloring of E(K n ).Erdős and Szekeres[5]proved that r(K k ) ≤ 4 k ≤ n, so this coloring contains a monochromatic K k .Together with the remaining n − k vertices, we obtain a monochromatic copy of H k,n .Proof of Lemma 2.2.Let N = nk, and consider the Turán coloring of E(K N ).Namely, we equitably partition V (K N ) into sets S 1 , . . ., S k , each of order n.We color all edges inside some S i red, and all edges between S i and S j blue, for all i = j.Since G k,n is connected and has n + 1 vertices, there can be no monochromatic red copy of G k,n , as the connected components of the red graph in this coloring each have n vertices.The blue graph, on the other hand, has chromatic number k, and G k,n has chromatic number k + 1, so there can be no monochromatic blue copy of G k,n either.This shows that r(G k,n ) > nk, as claimed.graph H is called d-degenerate if every subgraph of H has a vertex of degree at most d.Equivalently, H is d-degenerate if one can order its vertices so that every vertex has at most d neighbors which precede it in the ordering.
A The degeneracy of H is defined as the minimum d so that H is d-degenerate.Theorem 1.3 follows from the following result, which gives two bounds on r(G)/r(H) when H is obtained from G by deleting a vertex; the first is strong when H is sparse (i.e. has low degeneracy), while the second is stronger when H is dense (i.e. has high degeneracy).Balancing the two bounds, we obtain Theorem 1.3.Proposition 3.1.Let H be obtained from G by deleting a vertex.If H has n vertices and degeneracy d ≥ 1, then r(G) ≤ 4dr(H),