A natural generalisation in graph Ramsey theory

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 Erd\"os, Hajnal and Rado in 1965 and subsequently studied by Erd\"os and Szemer\'edi 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 characterise the class of the minimal $G_r$ in the case when $H$ is relaxed to be the graph property of containing a cycle; we then use it to progress towards a constructive description of that class by proving both a reduction and an extension theorem.Comment: 8 page

Generalised Ramsey numbers for two sets of cycles

We determine several generalised Ramsey numbers for two sets $\Gamma_1$ and $\Gamma_2$ of cycles, in particular, all generalised Ramsey numbers $R(\Gamma_1,\Gamma_2)$ such that $\Gamma_1$ or $\Gamma_2$ contains a cycle of length at most $6$, or the shortest cycle in each set is even. This generalises previous results of Erd\H{o}s, Faudree, Rosta, Rousseau, and Schelp from the 1970s. Notably, including both $C_3$ and $C_4$ in one of the sets, makes very little difference from including only $C_4$. Furthermore, we give a conjecture for the general case. We also describe many $(\Gamma_1,\Gamma_2)$-avoiding graphs, including a complete characterisation of most $(\Gamma_1,\Gamma_2)$-critical graphs, i.e., $(\Gamma_1,\Gamma_2)$-avoiding graphs on $R(\Gamma_1,\Gamma_2)-1$ vertices, such that $\Gamma_1$ or $\Gamma_2$ contains a cycle of length at most $5$. For length $4$, this is an easy extension of a recent result of Wu, Sun, and Radziszowski, in which $|\Gamma_1|=|\Gamma_2|=1$. For lengths $3$ and $5$, our results are new even in this special case. Keywords: generalised Ramsey number, critical graph, cycle, set of cyclesComment: 18 pages, 1 figur

Ramsey numbers of trees versus odd cycles

Burr, Erd\H{o}s, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum positive integer $n_0(m)$ such that this result holds for all $n \ge n_0(m)$, as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear. In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 50m$. Combining this with a result of Faudree, Lawrence, Parsons and Schelp yields $n_0(m)$ is bounded between two linear functions, thus identifying $n_0(m)$ up to a constant factor.Comment: 10 pages, updated to match EJC versio

Ramsey and Gallai-Ramsey number for wheels

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. Much like graph Ramsey numbers, Gallai-Ramsey numbers have gained a reputation as being very difficult to compute in general. As yet, still only precious few sharp results are known. In this paper, we obtain bounds on the Gallai-Ramsey number for wheels and the exact value for the wheel on $5$ vertices.Comment: arXiv admin note: text overlap with arXiv:1809.10298, arXiv:1902.1070

Ramsey numbers of ordered graphs under graph operations

An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph on $N$ vertices has a monochromatic copy of $\mathcal{G}$ that respects the ordering. In this paper we investigate the effect of various graph operations on the Ramsey number of a given ordered graph, and detail a general framework for applying results on extremal functions of 0-1 matrices to ordered Ramsey problems. We apply this method to give upper bounds on the Ramsey number of ordered matchings arising from sum-decomposable permutations, an alternating ordering of the cycle, and an alternating ordering of the tight hyperpath. We also construct ordered matchings on $n$ vertices whose Ramsey number is $n^{q+o(1)}$ for any given exponent $q\in(1,2)$

Planar Ramsey Numbers of Four Cycles Versus Wheels

For two given graphs $G$ and $H$ the planar Ramsey number $PR(G,H)$ is the smallest integer $n$ such that every planar graph $F$ on $n$ vertices either contains a copy of $G$, or its complement contains a copy of $H$. In this paper, we first characterize some structural properties of $C_4$-free planar graphs, and then we determine all planar Ramsey numbers $PR(C_4, W_n)$, for $n\ge 3$

Ramsey-type numbers involving graphs and hypergraphs with large girth

A question of Erd\H{o}s 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 Ruci\'nski. 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 $\textrm{girth}(H)=k$ and the Ramsey property that every $r$-colouring of $E(H)$ yields a monochromatic $C_k$. Two related numerical problems regarding arithmetic progressions in sets and cliques in graphs are also considered

On Induced Online Ramsey Number of Paths, Cycles, and Trees

An online Ramsey game is a game between Builder and Painter, alternating in turns. They are given a graph $H$ and a graph $G$ of an infinite set of independent vertices. In each round Builder draws an edge and Painter colors it either red or blue. Builder wins if after some finite round there is a monochromatic copy of the graph $H$, otherwise Painter wins. The online Ramsey number $\widetilde{r}(H)$ is the minimum number of rounds such that Builder can force a monochromatic copy of $H$ in $G$. This is an analogy to the size-Ramsey number $\overline{r}(H)$ defined as the minimum number such that there exists graph $G$ with $\overline{r}(H)$ edges where for any edge two-coloring $G$ contains a monochromatic copy of $H$. In this paper, we introduce the concept of induced online Ramsey numbers: the induced online Ramsey number $\widetilde{r}_{ind}(H)$ is the minimum number of rounds Builder can force an induced monochromatic copy of $H$ in $G$. We prove asymptotically tight bounds on the induced online Ramsey numbers of paths, cycles and two families of trees. Moreover, we provide a result analogous to Conlon [On-line Ramsey Numbers, SIAM J. Discr. Math. 2009], showing that there is an infinite family of trees $T_1,T_2,\dots$, $|T_i|<|T_{i+1}|$ for $i\ge1$, such that $$\lim_{i\to\infty} \frac{\widetilde{r}(T_i)}{\overline{r}(T_i)} = 0.$$Comment: 13 pages, 6 figure

Bipartite Ramsey numbers of large cycles

For an integer $r\geq 2$ and bipartite graphs $H_i$, where $1\leq i\leq r$, the bipartite Ramsey number $br(H_1,H_2,\ldots,H_r)$ is the minimum integer $N$ such that any $r$-edge coloring of the complete bipartite graph $K_{N,N}$ contains a monochromatic subgraph isomorphic to $H_i$ in color $i$ for some $i$, $1\leq i\leq r$. We show that for $\alpha_1,\alpha_2>0$, $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor})=(\alpha_1+\alpha_2+o(1))n$. We also show that if $r\geq 3, \alpha_1,\alpha_2>0, \alpha_{j+2}\geq [(j+2)!-1]\sum^{j+1}_{i=1} \alpha_i$ for $j=1,2,\ldots,r-2$, then $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor},\ldots,C_{2\lfloor \alpha_r n\rfloor})=(\sum^r_{j=1} \alpha_j+o(1))n.$ For $\xi>0$ and sufficiently large $n$, let $G$ be a bipartite graph with bipartition $\{V_1,V_2\}$, $|V_1|=|V_2|=N$, where $N=(2+8\xi)n$. We prove that if $\delta(G)>(\frac{7}{8}+9\xi)N$, then any $2$-edge coloring of $G$ contains a monochromatic copy of $C_{2n}$.Comment: 19 page

On ordered Ramsey numbers of bounded-degree graphs

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 $2$-coloring of the edges of the ordered complete graph on $N$ vertices contains a monochromatic copy of $\mathcal{G}$. We show that for every integer $d \geq 3$, almost every $d$-regular graph $G$ satisfies $\overline{R}(\mathcal{G}) \geq \frac{n^{3/2-1/d}}{4\log{n}\log{\log{n}}}$ for every ordering $\mathcal{G}$ of $G$. In particular, there are 3-regular graphs $G$ on $n$ vertices for which the numbers $\overline{R}(\mathcal{G})$ are superlinear in $n$, regardless of the ordering $\mathcal{G}$ of $G$. This solves a problem of Conlon, Fox, Lee, and Sudakov. On the other hand, we prove that every graph $G$ on $n$ vertices with maximum degree 2 admits an ordering $\mathcal{G}$ of $G$ such that $\overline{R}(\mathcal{G})$ is linear in $n$. We also show that almost every ordered matching $\mathcal{M}$ with $n$ vertices and with interval chromatic number two satisfies $\overline{R}(\mathcal{M}) \geq cn^2/\log^2{n}$ for some absolute constant $c$.Comment: 19 pages, 8 figures, minor correction