A simple graph-product type construction shows that for all natural numbers $r \ge q$, there exists an edge-coloring of the complete graph on $2^r$ vertices using $r$ colors where the graph consisting of the union of arbitrary $q$ color classes has chromatic number $2^q$. We show that for each fixed natural number $q$, if there exists an edge-coloring of the complete graph on $n$ vertices using $r$ colors where the graph consisting of the union of arbitrary $q$ color classes has chromatic number at most $2^q -1 $, then $n$ must be sub-exponential in $r$. This answers a question of Conlon, Fox, Lee, and Sudakov.
Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $H$. In this paper, we determine the Gallai-Ramsey numbers for connected graphs with five vertices of chromatic number four.
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.
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.
We construct bounded degree acyclic Borel graphs with large Borel chromatic number using a graph arising from Ramsey theory and limits of expander sequences.
Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of the complete graph $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $H$. In this paper, we study Gallai-Ramsey numbers for graphs with chromatic number three such as $\widehat{K}_m$ for $m\ge2$, where $\widehat{K}_m$ is a kipas with $m+1$ vertices obtained from the join of $K_1$ and $P_m$, and a class of graphs with five vertices, denoted by $\mathscr{H}$. We first study the general lower bound of s…
We apply Ramsey theoretic tools to show that there is a family of graphs which have tree-chromatic number at most~$2$ while the path-chromatic number is unbounded. This resolves a problem posed by Seymour.
In 1967, Erdős asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erdős and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3, t)$ shows that $f(n)$ is at most $(2 \sqrt{2} + o(1)) \sqrt{n/\log n}$. We improve this bound by a factor $\sqrt{2}$, as well as obtaining an analogous bound on the list chromatic number which is tight up to a constant factor. A bound in terms of the number of edges that is similarly tight follows, and these results confirm a conjecture of Cames van Batenburg, de Jo…
Mathematics Institute, University of Warwick（华威大学数学研究所）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极值组合与概率研究组）；Departament de Matemàtiques, Universitat Politècnica de Catalunya (UPC)（加泰罗尼亚理工大学数学系）；School of Mathematics and Statistics, Ningbo University（宁波大学数学与统计学院）
Given two graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any coloring of the edges of $K_N$ in red or blue yields a red $G$ or a blue $H$. Let $v(G)$ be the number of vertices of $G$ and $χ(G)$ be the chromatic number of $G$. Let $s(G)$ denote the chromatic surplus of $G$, the cardinality of a minimum color class taken over all proper colorings of $G$ with $χ(G)$ colors. Burr showed that for a connected graph $G$ and a graph $H$ with $v(G)\geq s(H)$, $R(G,H) \geq (v(G)-1)(χ(H)-1)+s(H)$. A connected graph $G$ is called $H$-good if $R(G,H)=(v(G)-1)(χ(H)-1)+s(…
The list Ramsey number $R_{\ell}(H,k)$, recently introduced by Alon, Bucić, Kalvari, Kuperwasser, and Szabó, is a list-coloring variant of the classical Ramsey number. They showed that if $H$ is a fixed $r$-uniform hypergraph that is not $r$-partite and the number of colors $k$ goes to infinity, $e^{Ω(\sqrt{k})} \le R_{\ell} (H,k) \le e^{O(k)}$. We prove that $R_{\ell}(H,k) = e^{Θ(k)}$ if and only if $H$ is not $r$-partite.
We consider unavoidable chromatic patterns in $2$-colorings of the edges of the complete graph. Several such problems are explored being a junction point between Ramsey theory, extremal graph theory (Turán type problems), zero-sum Ramsey theory, and interpolation theorems in graph theory. A role-model of these problems is the following: Let $G$ be a graph with $e(G)$ edges. We say that $G$ is omnitonal if there exists a function ${\rm ot}(n,G)$ such that the following holds true for $n$ sufficiently large: For any $2$-coloring $f: E(K_n) \to \{red, blue \}$ such that there are more than ${\rm …
For graphs G and H, let the induced Ramsey number IR(H,G) be the smallest number of vertices in a graph F such that any coloring of the edges of F in red and blue, there is either a red induced copy of H or a blue induced copy of G. In this note we consider the case when G=Sn is a star on n edges, for large n, and H is a fixed graph. We prove that (r-1)n < IR(H, Sn) < (r-1)(r-1)n + cn, for any c>0, sufficiently large n, and r denoting the chromatic number of H. The lower bound is asymptotically tight for any fixed bipartite H. The upper bound is attained up to a constant factor, for example by…
For a fixed infinite graph $H$, we study the largest density of a monochromatic subgraph isomorphic to $H$ that can be found in every two-coloring of the edges of $K_{\mathbb{N}}$. This is called the Ramsey upper density of $H$, and was introduced by Erdős and Galvin. Recently, the Ramsey upper density of the infinite path was determined. Here, we find the value of this density for all locally finite graphs $H$ up to a factor of 2, answering a question of DeBiasio and McKenney. We also find the exact density for a wide class of bipartite graphs, including all locally finite forests. Our approa…
We introduce the list colouring extension of classical Ramsey numbers. We investigate when the two Ramsey numbers are equal, and in general, how far apart they can be from each other. We find graph sequences where the two are equal and where they are far apart. For $\ell$-uniform cliques we prove that the list Ramsey number is bounded by an exponential function, while it is well-known that the Ramsey number is super-exponential for uniformity at least $3$. This is in great contrast to the graph case where we cannot even decide the question of equality for cliques.
The Ramsey number $R(F,H)$ is the minimum number $N$ such that any $N$-vertex graph either contains a copy of $F$ or its complement contains $H$. Burr in 1981 proved a pleasingly general result that for any graph $H$, provided $n$ is sufficiently large, a natural lower bound construction gives the correct Ramsey number involving cycles: $R(C_n,H)=(n-1)(χ(H)-1)+σ(H)$, where $σ(H)$ is the minimum possible size of a colour class in a $χ(H)$-colouring of $H$. Allen, Brightwell and Skokan conjectured that the same should be true already when $n\geq |H|χ(H)$. We improve this 40-year-old result of Bu…
A recent breakthrough of Conlon and Ferber yielded an exponential improvement on the lower bounds for multicolor diagonal Ramsey numbers. In this note, we modify their construction and obtain improved bounds for more than three colors.
Recently, Souza introduced blowup Ramsey numbers as a generalization of bipartite Ramsey numbers. For graphs $G$ and $H$, say $G\overset{r}{\longrightarrow} H$ if every $r$-edge-coloring of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. Then the blowup Ramsey number of $G,H,r,$ and $t$ is defined as the minimum $n$ such that $G[n] \overset{r}{\longrightarrow} H[t]$. Souza proved upper and lower bounds on $n$ that are exponential in $t$, and conjectured that the exponential constant does not depend on $G$. We prove that the dependence on $G$ in the exponentia…
This paper deals with the minimum number $m_H(r)$ of edges in an $H$-free graph with the chromatic number more than $r$. We show how bounds on Ramsey and Turán numbers imply bounds on $m_H(r)$.
Conventional Ramsey-theoretic investigations for edge-colourings of complete graphs are framed around avoidance of certain configurations. Motivated by considerations arising in the field of Qualitative Reasoning, we explore edge colourings that in addition to forbidding certain triangle configurations also require others to be present. These conditions have natural combinatorial interest in their own right, but also correspond to qualitative representability of certain nonassociative relation algebras, which we will call chromatic.
An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same…
