On the size-Ramsey number of grids

We show that the size-Ramsey number of the ? n ˆ ? n grid graph is O p n 5 { 4 q , improving a previous bound of n 3 { 2 ` o p 1 q by Clemens, Miralaei, Reding, Schacht, and Taraz.


Introduction
For graphs G and H, we say that G is Ramsey for H, and write G Ñ H, if every 2-colouring of the edges of G contains a monochromatic copy of H.In 1978, Erdős, Faudree, Rousseau, and Schelp [9] pioneered the study of the size-Ramsey number rpHq, defined as the smallest integer m for which there exists a graph G with m edges such that G Ñ H.The existence of the usual Ramsey number rpHq shows that this notion is sensible, since, for any H, it is easy to see that rpHq ď `rpHq 2 ˘.When H is a complete graph, this inequality is an equality, a simple fact first observed by Chvátal.
An early example showing that size-Ramsey numbers can exhibit interesting behaviour was found by Beck [1], who showed that P n , the path with n vertices, satisfies rpP n q " Opnq, which is significantly smaller than the Opn 2 q bound that follows from applying the inequality above and the corresponding bound rpP n q " Opnq for the usual Ramsey number of P n .In a follow-up paper, Beck [2] asked whether a similar phenomenon occurs for all bounded-degree graphs, that is, whether, for any integer ∆ ě 3, there exists a constant c such that any graph H with n vertices and maximum degree ∆ has size-Ramsey number at most cn.Although Rödl and Szemerédi [19] showed that this question has a negative answer already for ∆ " 3, much work has gone into extending Beck's result to other natural families of graphs, including: cycles [14], bounded-degree trees [10], powers of paths and bounded-degree trees [3,5,13], and more besides.
Most of the known families with linear size-Ramsey numbers have a bounded structural parameter, such as bandwidth [5] or, more generally, treewidth [15] (though see the recent papers [8,18] for examples with a somewhat different flavour).However, a fairly simple family of graphs which does not fall into any of these categories, but may still have linear size-Ramsey numbers, is the family of two-dimensional grid graphs.For s P N, the s ˆs grid is the graph with vertex set rss ˆrss where two pairs are adjacent if and only if they differ by one in exactly one coordinate.Obviously, the maximum degree of the s ˆs grid is four, but its bandwidth and treewidth are both exactly s (see, e.g., [4]), so the problem of estimating the size-Ramsey number of this graph, and usually we will take s " ?n so that the graph has n vertices, provides an interesting test case for exploring new ideas and techniques.
Regarding upper bounds for the size-Ramsey number of the ?n ˆ?n grid, an important result of Kohayakawa, Rödl, Schacht, and Szemerédi [17], which says that every graph H with n vertices and maximum degree ∆ satisfies rpHq ď n 2´1{∆`op1q , immediately yields the bound n 7{4`op1q .This was recently improved by Clemens, Miralaei, Reding, Schacht, and Taraz [6] to n 3{2`op1q (and an alternative proof of this bound was also noted in our recent paper [7]).The goal of this short note is to provide an elementary proof of an improved upper bound.
Theorem 1.1.There exists a constant C ą 0 such that the size-Ramsey number of the ?n ˆ?n grid graph is at most Cn 5{4 .
Like much of the work on size-Ramsey numbers, the previous bounds for grids were obtained by applying the sparse regularity method to show that every 2-colouring of the edges of the Erdős-Rényi random graph G n,p , for some appropriate density p, contains a monochromatic copy of the grid.However, it is a simple exercise in the first moment method to show that for p ! n ´1{2 the random graph G n,p with high probability does not contain the s ˆs grid graph as a subgraph if s " Θp ?nq, so the bound Opn 3{2 q is the best that one can hope to achieve using this procedure.
To see how it is that we gain on this bound, suppose that s " ?n.It is known [14] that there are K, ∆ ą 0 and a graph H with Ks vertices and maximum degree at most ∆ which is Ramsey for C s , the cycle of length s.Consider now a 'blow-up' Γ of H obtained by replacing every x P V pHq by an independent set V x of order Θpsq and every xy P H by a bipartite graph pV x , V y q in which every edge exists independently with probability p " Θps ´1{2 q.With high probability, such a blow-up contains Θps 5{2 q " Θpn 5{4 q edges.That is, instead of revealing a random graph G n,p on all n " Θps 2 q vertices, we only reveal edges that lie within Θpsq bipartite subgraphs, each with parts of order Θpsq.This salvages a significant number of edges which would otherwise go to waste.
Consider now a 2-colouring of Γ and recall that H was chosen so that H Ñ C s .A key lemma, Lemma 2.3 below, then allows us to conclude that there are sets V 1 , . . ., V s in Γ and a collection U i Ď V i of large subsets such that all pU i , U i`1 q with i P rss, where addition is taken modulo s, are 'regular' in the same colour.We may then sequentially embed the vertices of the grid so that the first row is embedded into U 1 , . . ., U s , the second into U 2 , . . ., U s , U 1 , and so on.

Definitions and key lemmas
In this section, we recall several standard definitions and note two key lemmas that will be needed in the proof of Theorem 1.1.Most of these revolve around the concept of sparse regularity (for a thorough overview of which we refer the reader to the survey by Gerke and Steger [12]).
For ε ą 0 and p P p0, 1s, a pair of sets pV 1 , V 2 q is said to be pε, pq-lower-regular in a graph G if, for all Immediately from this definition, we get that in every pε, pq-lower-regular pair pV 1 , V 2 q, for each i P t1, 2u, all but at most ε|V i | vertices in V i have degree at least p1 ´εqp|V 3´i | into V 3´i -a fact we will make use of in the proof of Theorem 1.1.Another useful and well-known property is that lower-regularity is inherited on large sets.Lemma 2.1.Let 0 ă ε ă δ, p P p0, 1s, and let pV 1 , V 2 q be an pε, pq-lower-regular pair.Then any pair of subsets V1 i Ď V i , i P t1, 2u, with |V 1 i | ě δ|V i | form an pε{δ, pq-lower-regular pair.
For λ ą 0 and p P p0, 1s, a graph G is said to be pλ, pq-uniform if, for all disjoint X, Y Ď V pGq with |X|, |Y | ě λ|V pGq|, the density of edges between X and Y satisfies d G pX, Y q " p1 ˘λqp.
If only the upper bound holds, the graph is said to be upper-uniform. 1 For example, it is easy to see that the random graph G n,p is with high probability pop1q, pq-uniform whenever p " 1{n.
If G " pV 1 , V 2 ; Eq is bipartite, we say that G is pλ, pq-uniform or upper-uniform if the same conditions hold for all In order to prove our main technical lemma, we rely on the following result, a simple corollary of [16,Lemma 6], whose proof follows a density increment argument.The same conclusion can also be obtained by an application of the sparse regularity lemma.
Lemma 2.2.For all 0 ă ε ă 1{2 and α P p0, 1q, there exists λ ą 0 such that the following holds for every p P p0, 1s.Let G " pV 1 , V 2 ; Eq be a pλ, pq-upper-uniform bipartite graph with The next lemma is the crux of our argument.Here and elsewhere, we say that pX, Y ; Eq is lower-regular if pX, Y q is lower-regular with respect to the set of edges E.
Lemma 2.3.For every r, ∆ ě 2 and ε ą 0, there exists λ ą 0 such that the following holds for every p P p0, 1s.Let H be a graph on at least two vertices with ∆pHq ď ∆ and let Γ be obtained by replacing every x P V pHq with an independent set V x of sufficiently large order n and every xy P H by a pλ, pq-uniform bipartite graph between V x and V y .Then, for every r-colouring of the edges of Γ, there exists an r-colouring ϕ of the edges of H and, for every x P V pHq, a subset U x Ď V x of order |U x | " λn such that pU x , U y ; E ϕpxyq q is pε, p{p2rqq-lower-regular for each xy P H, where E ϕpxyq Ď EpΓq stands for the edges in colour ϕpxyq.
Proof.Given ε, r, and ∆, we let α " 1{p2rq, ε ∆`1 :" ε, λ ∆`1 " λ 2.2 pε ∆`1 , αq, and, for every i " ∆, . . ., 1, sequentially take ε i " ε i`1 λ i`1 and λ i " λ 2.2 pε i , αq.Lastly, let λ " ś iPr∆`1s λ i .Fix any r-colouring of (the edges of) Γ and, for every c P rrs, let Γ c stand for the subgraph (in terms of edges) in colour c.Note that H has edge-chromatic number at most ∆ `1.In other words, there exists a partition of the edges of H into H 1 , . . ., H ∆`1 such that each H i is a matching.We find the required collection tU x u xPV pHq by maintaining the following condition for every i P r∆ `1s: for every x P V pHq, there exists a chain x | for all j P ris and (ii ) for every xy P Ť jďi H j , pU i x , U i y q is pε i , αpq-lower-regular in Γ c for some c P rrs.
Consequently, for i " ∆ `1, we obtain sets U x Ď V x , for every x P V pHq, of order |U x | " p ś iPr∆`1s λ i qn " λn such that pU x , U y q is pε ∆`1 , αpq-lower-regular and, thus, pε, αpq-lowerregular for every xy P H.It remains to show that we can indeed do this.Consider first i " 1.For each xy P H 1 , let c P rrs be the majority colour in ΓrV x , V y s.As e Γc pV x , V y q ě p1 ´λqn 2 p{r, we may apply Lemma 2.2 with ε 1 (as ε) and Γ c rV x , V y s (as G) to obtain sets U 1 x , U 1 y with the desired properties.For every x P V pHq which is isolated in H 1 , we simply take an arbitrary subset U 1 x Ď V x of order λ 1 |U 0 x |.Thus, the required condition holds for i " 1.
Suppose now that the condition holds for some i ě 1 and let us show that it also holds for i `1.As above, for every xy P H i`1 , let c P rrs be the majority colour in ΓrV x , V y s.Since ΓrV x , V y s is pλ, pq-uniform and, by (i ), |U i x |, |U i y | ě λn, we have e Γ pU i x , U i y q " p1 ˘λq|U i x ||U i y |p and, hence, for which pU i`1 x , U i`1 y q is pε i`1 , αpq-lower-regular in Γ c .For every x P V pHq which is isolated in H i`1 , we again take an arbitrary subset Observe also that, for every xz P Ť jďi H j , since pU i x , U i z q was pε i , αpq-lower-regular in Γ c 1 for some c 1 P rrs and |U i`1 x | " λ i`1 |U i x |, Lemma 2.1 and the fact that ε i {λ i`1 " ε i`1 imply that pU i`1 x , U i`1 z q is pε i`1 , αpq-lower-regular in Γ c 1 , as desired.This completes the proof.
We also need a variant of a result from our previous paper [7, Lemma 3.5] about regularity inheritance.While that result was stated for the usual (full) notion of regularity, we only need lower-regularity here, allowing us to save a factor of plog nq 1{2 .Lemma 2.4.For all ε, α, λ ą 0, there exist positive constants ε 1 pε, αq and Cpε, α, λq such that for p ě Cn ´1{2 , with probability at least 1 ´opn ´5q, the random graph Γ " G n,p has the following property.
Suppose G Ď Γ and V 1 , V 2 Ď V pΓq are disjoint subsets of order ñ " λn such that pV 1 , V 2 q is pε 1 , αpq-lower-regular in G. Then there exists B Ď V pΓq of order |B| ď εñ such that, for each v, w P V pΓq pV 1 Y V 2 Y Bq (not necessarily distinct), the following holds: for any two subsets N v Ď N Γ pv, V 1 q and N w Ď N Γ pw, V 2 q of order αñp{4, both pN v , V 2 q and pN v , N w q are pε, αpq-lower-regular in G.
Sketch of the proof.The proof proceeds along the same lines as the proof of [7,Lemma 3.5].The only difference is that there we made use of an inheritance lemma for full regularity (namely, Corollary 3.5 in [20]), which requires the sets on which regularity is inherited to be of order at least C log n{p, resulting in the requirement that p ě Cplog n{nq 1{2 .However, for lower-regularity, one can instead use the inheritance lemma of Gerke, Kohayakawa, Rödl, and Steger [11, Corollary 3.8], which only requires the sets to be of order at least C{p, resulting in p ě Cn ´1{2 .The rest of the proof remains exactly the same.

Proof of Theorem 1.1
Since it requires no additional work, we will actually prove the r-colour analogue of Theorem 1.1.More precisely, we will show that for every integer r ě 2 there exists a graph of order n with Opn 5{4 ) edges for which every r-colouring of the edges contains a monochromatic copy of the δ ?n ˆδ? n grid for some δ ą 0.
We show that the size-Ramsey number of the δs ˆδs grid is Ops 5{2 q, which, for s " ?n, implies the desired statement.
Let Γ be a graph obtained by replacing every vertex x P V pHq by an independent set V x of order s and every edge xy P H by a bipartite graph between V x and V y in which each edge exists independently with probability p " Cs ´1{2 for some sufficiently large constant C ą 0. With high probability, Γ has the following property: This is a standard feature of random graphs and follows from the Chernoff bound together with an application of the union bound.In particular, it establishes that with high probability ΓrV x , V y s is pλ, pq-uniform for every xy P H and, therefore, Γ has at most Ks ¨∆{2 ¨p1 `λqs2 p " Ops 5{2 q edges.Additionally, with high probability, Γ is such that every ΓrV x Y V y Y V z s has the property of Lemma 2.4 (applied with ε{9 as ε, λ{3 as λ, and 3s as n) for every path xyz of length two in H. 2 This again follows from the union bound, as there are Opsq such paths in total and the conclusion of Lemma 2.4 holds with probability 1 ´ops ´5q for every fixed path.We now fix an outcome of Γ which satisfies all of these properties.
Consider some r-colouring of the edges of Γ and let ϕ be the colouring of the edges of H given by Lemma 2.3 (applied with ε 1 as ε).By the choice of H, this colouring contains a monochromatic copy of C δs , which, without loss of generality, we may assume has vertices 1, . . ., δs.Therefore, there is a colour c P rrs and sets U i of order s " λs in Γ such that, for every i P rδss, the pair pU i , U i`1 q is pε 1 , αpq-lower-regular in the subgraph of Γ induced by colour c, where we identify δs `i with i.Let G be the graph induced by these sets whose edges are the edges of Γ of colour c.We will show that G contains the δs ˆδs grid as a subgraph.
For every i P rδss, let B Ď U i Y U i`1 Y U i`2 be the set given by Lemma 2.4 (which was applied with ε{9 as ε, λ{3 as λ, and 3s as n) on ΓrU i Y U i`1 Y U i`2 s, which is a set of 'bad vertices' for the pair pU i`1 , U i`2 q.As each U i is a part of three such applications, by the chosen properties of Γ, for every i P rδss there exists a set B i Ď U i of order |B i | ď εs such that: of order αsp{4 and Our plan is to embed the vertex pi, jq of the δs ˆδs grid into U i`j´1 .The next claim helps us achieve this.
Proof of Claim 3.1.Without loss of generality, we may assume that all the Q i`j´1 are of order 2εs, as we can take arbitrary supersets if this is not the case.Let S 1 i`j´1 Ď S i`j´1 be the set of all v P S i`j´1 with at least αsp{4 neighbours in U i`j Q i`j .On the one hand, as pS i`j´1 , U i`j B i`j q is pε, αpq-lower-regular and, thus, there are fewer than ε|S i`j´1 | vertices in S i`j´1 with degree less than αsp{2 in U i`j B i`j , we have On the other hand, assuming S i`j´1 S 1 i`j´1 is of order at least αsp{16 and, hence, for C ą 0 sufficiently large, property (A1) implies that e G pS i`j´1 S 1 i`j´1 , Q i`j q ď p1 `λq2εs|S i`j´1 S 1 i`j´1 |p.
Since ε ă α{128, this is a contradiction.Therefore, there are sets S 1 i`j´1 Ď S i`j´1 of order at least |S i`j´1 |´αsp{16 for each j P rδss such that every v P S 1 i`j´1 satisfies |N G pv, U i`j Q i`j q| ě αsp{4.
We declare the set of such vertices to be S 2 i`j´2 and continue on to the next index j.Starting with an arbitrary v 1 P S 2 i and sequentially choosing v j P N G pv j´1 , S 2 i`j´1 q now completes the proof.