For an integer $q\ge 2$, a graph $G$ is called $q$-Ramsey for a graph $H$ if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. If $G$ is $q$-Ramsey for $H$, yet no proper subgraph of $G$ has this property then $G$ is called $q$-Ramsey-minimal for $H$. Generalising a statement by Burr, Nešetřil and Rödl from 1977 we prove that, for $q\ge 3$, if $G$ is a graph that is not $q$-Ramsey for some graph $H$ then $G$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$, as long as $H$ is $3$-connected or isomorphic to the triangle. For such $H$, the following are some consequences.
(1) For $2\le r< q$, every $r$-Ramsey-minimal graph for $H$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$.
(2) For every $q\ge 3$, there are $q$-Ramsey-minimal graphs for $H$ of arbitrarily large maximum degree, genus, and chromatic number.
(3) The collection $\{{\cal M}_q(H) : H \text{ is 3-connected or } K_3\}$ forms an antichain with respect to the subset relation, where ${\cal M}_q(H)$ denotes the set of all graphs that are $q$-Ramsey-minimal for $H$.
We also address the question which pairs of graphs satisfy ${\cal M}_q(H_1)={\cal M}_q(H_2)$, in which case $H_1$ and $H_2$ are called $q$-equivalent. We show that two graphs $H_1$ and $H_2$ are $q$-equivalent for even $q$ if they are $2$-equivalent, and that in general $q$-equivalence for some $q\ge 3$ does not necessarily imply $2$-equivalence. Finally we indicate that for connected graphs this implication may hold: Results by Nešetřil and Rödl and by Fox, Grinshpun, Liebenau, Person and Szabó imply that the complete graph is not $2$-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.
The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $n$ for which every $k$-edge-coloured complete bipartite graph $K_{n,n}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and, independently, by Gyárfás and Lehel, who determined the $2$-colour Ramsey number of paths. In this paper we determine asymptotically the $3$-colour bipartite Ramsey number of paths and (even) cycles.
We estimate the $3$-colour bipartite Ramsey number for balanced bipartite graphs $H$ with small bandwidth and bounded maximum degree. More precisely, we show that the minimum value of $N$ such that in any $3$-edge colouring of $K_{N,N}$ there is a monochromatic copy of $H$ is at most $\big(3/2+o(1)\big)|V(H)|$. In particular, we determine asymptotically the $3$-colour bipartite Ramsey number for balanced grid graphs.
The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $N$ for which every $k$-edge-coloured complete bipartite graph $K_{N,N}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was initiated over 40 years ago by Faudree and Schelp and, independently, by Gyárfás and Lehel, who determined the $2$-colour bipartite Ramsey number of paths. Recently the $3$-colour Ramsey number of paths and (even) cycles, was essentially determined as well. Improving the results of DeBiasio, Gyárfás, Krueger, Ruszinkó, and Sárközy, in this paper we determine a…
In 2015 Bloom and Liebenau proved that $K_n$ and $K_n+K_{n-1}$ possess the same $2$-Ramsey graphs for all $n\geq 3$ (with a single exception for $n=3$). In the following we give a simple proof that $K_n$ and $K_n+K_{n-1}$ possess the same $r$-Ramsey graphs for all $n, r\geq 3$.
We exhibit a family of $3$-uniform hypergraphs with the property that their $2$-colour Ramsey numbers grow polynomially in the number of vertices, while their $4$-colour Ramsey numbers grow exponentially. This is the first example of a class of hypergraphs whose Ramsey numbers show a strong dependence on the number of colours.
We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a {\em set} of colours (instead of just one colour). We give bounds for monochromatic tree covers in this setting, both for an underlying complete graph, and an underlying complete bipartite graph. We also discuss a generalisation of Ramsey numbers to our setting and propose some other new directions. Our results for tree covers in complete graphs imply that a stronger version of Ryser's conjecture holds for $k$-intersecting $r$-partite $r$-unif…
In this paper we prove a new result about partitioning coloured complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for k at least 1, in every edge colouring of a complete graph with the colours red and blue, it is possible to cover all the vertices with k disjoint red paths and a disjoint blue balanced complete (k+1)-partite graph. When the colouring is connected in red, we prove a stronger result - that it is possible to cover all the vertices with k red paths and a blue balanced complete (k+2)-partite graph. Using these results w…
Given a positive integer $s$, the $s$-colour size-Ramsey number of a graph $H$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that, in any colouring of $E(G)$ with $s$ colours, there is a monochromatic copy of $H$. We prove that, for any positive integers $k$ and $s$, the $s$-colour size-Ramsey number of the $k$th power of any $n$-vertex bounded degree tree is linear in $n$. As a corollary we obtain that the $s$-colour size-Ramsey number of $n$-vertex graphs with bounded treewidth and bounded degree is linear in $n$, which answers a question rai…
The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$ such that if each edge of the complete graph $K_n$ is assigned a set of $s$ colours from $\{1,\ldots,r\}$, then one of the colours contains a monochromatic clique of size $k$. The case $s = 1$ is the usual $r$-colour Ramsey number, and the case $s = r - 1$ was studied by Erdős, Hajnal and Rado in 1965, and by Erdős and Szemerédi in 1972. The first significant results for general $s$ were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraëte, who showed that $R_{r,s}(k) = 2^{Θ(kr)}$ if $s/r$ is bounde…
The two-colour Ramsey number $R(m,n)$ is the least natural number $p$ such that any graph of order $p$ must contain either a clique of size $m$ or an independent set of size $n$. We exhibit a method for computing upper bounds for $R(m,n)$ recursively, using known upper bounds of $R(\cdot,\cdot)$ with lower values for at least one of the arguments. We also give an example of how this method could be used to improve several of the best known bounds that are available in the literature (which however soon will be obsolete due to a forthcoming work).
The anti-Ramsey number $AR(n,G$), for a graph $G$ and an integer $n\geq|V(G)|$, is defined to be the minimal integer $r$ such that in any edge-colouring of $K_n$ by at least $r$ colours there is a multicoloured copy of $G$, namely, a copy of $G$ whose edges have distinct colours. In this paper we determine the anti-Ramsey numbers of all graphs having at most four edges.
The grid Ramsey number $ G(r) $ is the smallest number $ n $ such that every edge-colouring of the grid graph $Γ_{n,n} := K_n \times K_n$ with $r$ colours induces a rectangle whose parallel edges receive the same colour. We show $ G(r) \leq r^{\binom{r+1}{2}} - \left( 1/4 - o(1) \right) r^{\binom{r}{2}+1} $, slightly improving the currently best known upper bound due to Gyárfás.
Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and show that either one can find an $m$-coloured complete subgraph for every natural number $m$ or there exists an infinite subset $X \subset \mathbb{N}$ coloured in one of two canonical ways: either the colouring is injective on $X$ or there exists a distinguished vertex $v$ in $X$ such that $X \setminus \lbrace v \rbrace$ is $1$-coloured and each edge betwee…
In this paper, we study Ramsey-type problems for directed graphs. We first consider the $k$-colour oriented Ramsey number of $H$, denoted by $\overrightarrow{R}(H,k)$, which is the least $n$ for which every $k$-edge-coloured tournament on $n$ vertices contains a monochromatic copy of $H$. We prove that $ \overrightarrow{R}(T,k) \le c_k|T|^k$ for any oriented tree $T$. This is a generalisation of a similar result for directed paths by Chvátal and by Gyárfás and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor. We also consider the $k$-colour directed Rams…
In 1991, McKay and Radziszowski proved that, however each 3-subset of a 13-set is assigned one of two colours, there is some 4-subset whose four 3-subsets have the same colour. More than 25 years later, this remains the only non-trivial classical Ramsey number known for hypergraphs. In this article, we find all the extremal colourings of the 3-subsets of a 12-set and list some of their properties. Using the catalogue, we provide an answer to a question of Dudek, Fleur, Mubayi and Rödl about the size-Ramsey numbers of hypergraphs.
A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let s_r(H) denote the smallest minimum degree of G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter s_2 was initiated by Burr, Erdős, and Lovász in 1976 when they showed that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of s_r(K_k) on r and show that, under the condition that k …
A graph $G$ is Ramsey for a graph $H$ if every colouring of the edges of $G$ in two colours contains a monochromatic copy of $H$. Two graphs $H_1$ and $H_2$ are Ramsey equivalent if any graph $G$ is Ramsey for $H_1$ if and only if it is Ramsey for $H_2$. A graph parameter $s$ is Ramsey distinguishing if $s(H_1)\neq s(H_2)$ implies that $H_1$ and $H_2$ are not Ramsey equivalent. In this paper we show that the chromatic number is a Ramsey distinguishing parameter. We also extend this to the multi-colour case and use a similar idea to find another graph parameter which is Ramsey distinguishing.
In this note we study graphs $G_r$ with the property that every colouring of $E(G_r)$ with $r+1$ colours admits a copy of some graph $H$ using at most $r$ colours. For $1\le r\le e(H)$ such graphs occur naturally at intermediate steps in the synthesis of a $2$-colour Ramsey graph $G_1\longrightarrow H$. (The corresponding notion of Ramsey-type numbers was introduced by Erdös, Hajnal and Rado in 1965 and subsequently studied by Erdös and Szemerédi in 1972). For $H=K_n$ we prove a result on building a $G_{r}$ from a $G_{r+1}$ and establish Ramsey-infiniteness. From the structural point of view, …
We show that for any positive integer $r$ there exists an integer $k$ and a $k$-colouring of the edges of $K_{2^{k}+1}$ with no monochromatic odd cycle of length less than $r$. This makes progress on a problem of Erdős and Graham and answers a question of Chung. We use these colourings to give new lower bounds on the $k$-colour Ramsey number of the odd cycle and prove that, for all odd $r$ and all $k$ sufficiently large, there exists a constant $ε= ε(r) > 0$ such that $R_{k}(C_{r}) > (r-1)(2+ε)^{k-1}$.
It was previously shown that any two-colour colouring of K(C_n) must contain a monochromatic planar K_4 subgraph for n >= N^*, where 6 <= N^* <= N and N is Graham's number. The bound was later improved to 11 <= N^* <= N. In this article, it is improved to 13 <= N^* <= N.
Let $P_t$ denote the path on $t$ vertices. The $r$-coloured Ramsey number of $P_t$, denoted by $R_r(P_t)$, is the minimum integer $n$ such that whenever the complete graph on $n$ vertices is given an $r$-edge-colouring, there exists a monochromatic copy of $P_t$. In this note, we determine $R_r(P_5)$, which is approximately $3r$.
Given bipartite graphs $G_1, \ldots, G_n$, the bipartite Ramsey number $BR(G_1,\ldots, G_n)$ is the last integer $b$ such that any complete bipartite graph $K_{b,b}$ with edges coloured with colours $1,2,\ldots,n$ contains a copy of some $G_i$ ($1\leq i\leq n$) where all edges of $G_i$ have colour $i$. As another view of bipartite Ramsey numbers, for given bipartite graphs $G_1, \ldots, G_n$ and a positive integer $m$, the $m$-bipartite Ramsey number $BR_m(G_1, \ldots, G_n)$, is defined as the least integer $b$, such that any complete bipartite graph $K_{m,b}$ with edges coloured with colours …
The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and Rödl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and powers of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers tha…
We study Ramsey's theorem for pairs and two colours in the context of the theory of $α$-large sets introduced by Ketonen and Solovay. We prove that any $2$-colouring of pairs from an $ω^{300n}$-large set admits an $ω^n$-large homogeneous set. We explain how a formalized version of this bound gives a more direct proof, and a strengthening, of the recent result of Patey and Yokoyama [Adv. Math. 330 (2018), 1034--1070] stating that Ramsey's theorem for pairs and two colours is $\forallΣ^0_2$-conservative over the axiomatic theory $\mathsf{RCA}_0$ (recursive comprehension).
We prove that, for $n\geq 4$, the graphs $K_n$ and $K_n+K_{n-1}$ are Ramsey equivalent. That is, if $G$ is such that any red-blue colouring of its edges creates a monochromatic $K_n$ then it must also possess a monochromatic $K_n+K_{n-1}$.
The following relaxation of the classical problem of determining Ramsey number of a fixed graph has first been proposed by Erdos, Hajnal and Rado over 50 years ago. Given a graph $G$ and an integer $t \geq 2$ determine the minimum number $N$ such that in any $t$-coloured complete graph on $N$ vertices there is a copy of $G$ using only edges of some $t-1$ colours. We determine the answer precisely when $G$ is a path.
For a graph $L$ and an integer $k\geq 2$, $R_k(L)$ denotes the smallest integer $N$ for which for any edge-colouring of the complete graph $K_N$ by $k$ colours there exists a colour $i$ for which the corresponding colour class contains $L$ as a subgraph. Bondy and Erdős conjectured that for an odd cycle $C_n$ on $n$ vertices, $$R_k(C_n) = 2^{k-1}(n-1)+1 \text{for $n>3$.}$$ They proved the case when $k=2$ and also provided an upper bound $R_k(C_n)\leq (k+2)!n$. Recently, this conjecture has been verified for $k=3$ if $n$ is large. In this note, we prove that for every integer $k\geq 4$, $$R_k(C…
A question of Erdős asks if for every pair of positive integers $r$ and $k$, there exists a graph $H$ having $\textrm{girth}(H)=k$ and the property that every $r$-colouring of the edges of $H$ yields a monochromatic cycle $C_k$. The existence of such graphs was confirmed by the third author and Ruciński. We consider the related numerical problem of determining the smallest such graph with this property. We show that for integers $r$ and $k$, there exists a graph $H$ on $R^{10k^2} k^{15k^3}$ vertices (where $R = R(C_k;r)$ is the $r$-colour Ramsey number for the cycle $C_k$) having $\textrm{girt…
We prove that for all epsilon>0 there are c>0 and n_0 such that for all n>n_0 the following holds. For any two-colouring of the edges of $K_{n,n,n}$ one colour contains copies of all trees T of order t<(3-epsilon)n/2 and with maximum degree at most n^c. This confirms a conjecture of Schelp.
For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show that for fixed $k\geq2$ and $n$ odd and sufficiently large, \[ R_k(C_n)=2^{k-1}(n-1)+1. \] This resolves a conjecture of Bondy and Erdős [J. Combin. Th. Ser. B \textbf{14} (1973), 46--54] for large $n$. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimis…
The size-Ramsey number $\hat{R}(F,r)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with $r$ colours yields a monochromatic copy of $F$. In this short note, we give an alternative proof of the recent result of Krivelevich that $\hat{R}(P_n,r) = O((\log r)r^2 n)$. This upper bound is nearly optimal, since it is also known that $\hat{R}(P_n,r) = Ω(r^2 n)$.
Given a positive integer $s$, a graph $G$ is $s$-Ramsey for a graph $H$, denoted $G\rightarrow (H)_s$, if every $s$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The $s$-colour size-Ramsey number ${\hat{r}}_s(H)$ of a graph $H$ is defined to be ${\hat{r}}_s(H)=\min\{|E(G)|\colon G\rightarrow (H)_s\}$. We prove that, for all positive integers $k$ and $s$, we have ${\hat{r}}_s(P_n^k)=O(n)$, where $P_n^k$ is the $k$th power of the $n$-vertex path $P_n$.
The size-Ramsey number $\hat{R}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In this paper, first we focus on the size-Ramsey number of a path $P_n$ on $n$ vertices. In particular, we show that $5n/2-15/2 \le \hat{R}(P_n) \le 74n$ for $n$ sufficiently large. (The upper bound uses expansion properties of random $d$-regular graphs.) This improves the previous lower bound, $\hat{R}(P_n) \ge (1+\sqrt{2})n-O(1)$, due to Bollobás, and the upp…
We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant $C$ such that \[r(k+1, k+1) \leq k^{- C \frac{\log k}{\log \log k}} \binom{2k}{k}.\]
A well-known result of Rödl and Ruciński states that for any graph $H$ there exists a constant $C$ such that if $p \geq C n^{- 1/m_2(H)}$, then the random graph $G_{n,p}$ is a.a.s. $H$-Ramsey, that is, any $2$-colouring of its edges contains a monochromatic copy of $H$. Aside from a few simple exceptions, the corresponding $0$-statement also holds, that is, there exists $c>0$ such that whenever $p\leq cn^{-1/m_2(H)}$ the random graph $G_{n,p}$ is a.a.s. not $H$-Ramsey. We show that near this threshold, even when $G_{n,p}$ is not $H$-Ramsey, it is often extremely close to being $H$-Ramsey. More…
A relational structure $\mathrm{R}$ is {\em rainbow Ramsey} if for every finite induced substructure $\mathrm{C}$ of $\mathrm{R}$ and every colouring of the copies of $\mathrm{C}$ with countably many colours, such that each colour is used at most $k$ times for a fixed $k$, there exists a copy $\mathrm{R}^\ast$ of $\mathrm{R}$ so that the copies of $\mathrm{C}$ in $\mathrm{R^\ast}$ use each colour at most once. We show that certain ultrahomogenous binary relational structures, for example the Rado graph, are rainbow Ramsey. Via compactness this then implies that for all finite graphs $\mathrm{B…
We show that every two-colouring of the edges of the complete graph $K_n$ contains a monochromatic trail or circuit of length at least $2n^2/9 +o(n^2)$, which is asymptotically best possible.
We show that for every finite colouring of the natural numbers there exists $a,b >1$ such that the triple $\{a,b,a^b\}$ is monochromatic. We go on to show the partition regularity of a much richer class of patterns involving exponentiation. For example, as a corollary to our main theorem, we show that for every $n \in \mathbb{N}$ and for every finite colouring of the natural numbers, we may find a monochromatic set including the integers $x_1,\ldots,x_n >1$; all products of distinct $x_i$; and all "exponential compositions" of distinct $x_i$ which respect the order $x_1,\ldots,x_n$. In particu…
We define a collection of topological Ramsey spaces consisting of equivalence relations on $ω$ with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of $ω$. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) i…
Given a hypergraph $H$, the size-Ramsey number $\hat{r}_2(H)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that in any colouring of the edges of $G$ with two colours there is a monochromatic copy of $H$. We prove that the size-Ramsey number of the $3$-uniform tight path on $n$ vertices $P^{(3)}_n$ is linear in $n$, i.e., $\hat{r}_2(P^{(3)}_n) = O(n)$. This answers a question by Dudek, Fleur, Mubayi, and Rödl for $3$-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved $\hat{r}_2(P^{(3)}…
The size Ramsey number $\hat{r}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In 1983, Beck provided a beautiful argument that shows that $\hat{r}(P_n)$ is linear, solving a problem of Erdős. In this short note, we provide an alternative but elementary proof of this fact that actually gives a better bound, namely, $\hat{r}(P_n) < 137n$ for $n$ sufficiently large.
Answering a question raised by Dudek and Prałat, we show that if $pn\rightarrow \infty$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there exists a monochromatic path of length $n(2/3+o(1))$. This result is optimal in the sense that $2/3$ cannot be replaced by a larger constant. As part of the proof we obtain the following result which may be of independent interest. We show that given a graph $G$ on $n$ vertices with at least $(1-ε)\binom{n}{2}$ edges, whenever $G$ is $2$-edge-coloured, there is a monochromatic path of length at least $(2/3-100\sqrtε)n$. This is an extension of the classical…
We determine the Ramsey number of a connected clique matching. That is, we show that if $G$ is a $2$-edge-coloured complete graph on $(r^2 - r - 1)n - r + 1$ vertices, then there is a monochromatic connected subgraph containing $n$ disjoint copies of $K_r$, and that this number of vertices cannot be reduced.
Denote by $R(G_1, G_2, G_3)$ the minimum integer $N$ such that any three-colouring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of a graph $G_i$ coloured with colour $i$ for some $i\in{1,2,3}$. In a series of three papers of which this is the third, we consider the case where $G_1, G_2$ and $G_3$ are cycles of mixed parity. Specifically, in this in this paper, we consider $R(C_n,C_m,C_{\ell})$, where $n$ is even and $m$ and $\ell$ are odd. Figaj and Łuczak determined an asymptotic result for this case, which we improve upon to give an exact result. We prove …
We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\] The old bound, due to Erdős and Hajnal, was \[r_3 (l,l,l) \geq 2^{c l^2 \log^2 l}.\]
School of Mathematics and Statistics, University of Melbourne（墨尔本大学数学与统计学院）；Department of Mathematics and Statistics, University of Victoria（维多利亚大学数学与统计系）
