On some three color Ramsey numbers for paths, cycles, stripes and stars

For given graphs $G_{1}, G_{2}, ... , G_{k}, k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, ... , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with $k$ colors, then it always contains a monochromatic copy of $G_{i}$ colored with $i$, for some $1 \leq i \leq k$. The bipartite Ramsey number $b(G_1, \cdots, G_k)$ is the least positive integer $b$ such that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a monochromatic copy of bipartite $G_i$ in the $i$-th color, for some $i$, $1 \le i \le k$. There is very little known about $R(G_{1},\ldots, G_{k})$ even for very special graphs, there are a lot of open cases. In this paper, by using bipartite Ramsey numbers we obtain the exact values of some multicolor Ramsey numbers. We show that for sufficiently large $n_{0}$ and three following cases: 1. $n_{1}=2s$, $n_{2}=2m$ and $m-1<2s$, 2. $n_{1}=n_{2}=2s$, 3. $n_{1}=2s+1$, $n_{2}=2m$ and $s<m-1<2s+1$, we have $$R(C_{n_0}, P_{n_{1}},P_{n_{2}}) = n_0 + \Big \lfloor \frac{n_1}{2} \Big \rfloor + \Big \lfloor \frac{n_2}{2} \Big \rfloor -2.$$ We prove that $R(P_n,kK_{2},kK_{2})=n+2k-2$ for large $n$. In addition, we prove that for even $k$, $R((k-1)K_{2},P_{k},P_{k})=3k-4$. For $s < m-1<2s+1$ and $t\geq m+s-1$, we obtain that $R(tK_{2},P_{2s+1},P_{2m})=s+m+2t-2$ where $P_{k}$ is a path on $k$ vertices and $tK_{2}$ is a matching of size $t$. We also provide some new exact values or generalize known results for other multicolor Ramsey numbers of paths, cycles, stripes and stars versus other graphs

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

A survey of hypergraph Ramsey problems

The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We survey a variety of problems and results in hypergraph Ramsey theory that have grown out of understanding the quantitative aspects of $r_k(s,n)$. Our focus is on recent developments and open problems

Restricted size Ramsey number for P3P_3 versus cycles

Let $F$, $G$ and $H$ be simple graphs. We say $F \rightarrow (G, H)$ if for every $2$-coloring of the edges of $F$ there exists a monochromatic $G$ or $H$ in $F$. The Ramsey number $r(G, H)$ is defined as $r(G, H) = min\{|V (F)|: F \rightarrow (G, H)\}$, while the restricted size Ramsey number $r^{*}(G, H)$ is defined as $r^{*}(G, H) = min\{|E (F)|: F \rightarrow (G, H) , |V (F) | = r(G, H)\}$. In this paper we determine previously unknown restricted size Ramsey numbers $r^{*}(P_3, C_n)$ for $7 \leq n \leq 12$. We also give new upper bound $r^{*}(P_3, C_n) \leq 2n-2$ for even $n \geq 8$

Trees and nn-Good Hypergraphs

Trees fill many extremal roles in graph theory, being minimally connected and serving a critical role in the definition of $n$-good graphs. In this article, we consider the generalization of trees to the setting of $r$-uniform hypergraphs and how one may extend the notion of $n$-good graphs to this setting. We prove numerous bounds for $r$-uniform hypergraph Ramsey numbers involving trees and complete hypergraphs and show that in the $3$-uniform case, all trees are $n$-good when $n$ is odd or $n$ falls into specified even cases.Comment: 23 pages, 3 figures, 2 table

On the size Ramsey number of all cycles versus a path

We say $G\to (\mathcal{C}, P_n)$ if $G-E(F)$ contains an $n$-vertex path $P_n$ for any spanning forest $F\subset G$. The size Ramsey number $\hat{R}(\mathcal{C}, P_n)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges for which $G\to (\mathcal{C}, P_n)$. Dudek, Khoeini and Pra{\l}at proved that for sufficiently large $n$, $2.0036n \le \hat{R}(\mathcal{C}, P_n)\le 31n$. In this note, we improve both the lower and upper bounds to $2.066n\le \hat{R}(\mathcal{C}, P_n)\le 5.25n+O(1).$ Our construction for the upper bound is completely different than the one considered by Dudek, Khoeini and Pra{\l}at. We also have a computer assisted proof of the upper bound $\hat{R}(\mathcal{C}, P_n)\le \frac{75}{19}n +O(1) < 3.947n$.Comment: 13 pages, 3 figure

A Note on Lower Bounds for Induced Ramsey Numbers

We say that a graph $F$ strongly arrows a pair of graphs $(G,H)$ if any 2-colouring of its edges with red and blue leads to either a red $G$ or a blue $H$ appearing as induced subgraphs of $F$. The induced Ramsey number, $IR(G,H)$ is defined as the minimum number of vertices of a graph $F$ which strongly arrows a pair $(G,H)$. We will consider two aspects of induced Ramsey numbers. Firstly there will be shown that the lower bound of the induced Ramsey number for a connected graph $G$ with independence number $\alpha$ and a graph $H$ with clique number $\omega$ roughly $\frac{\omega^2\alpha}{2}$. This bounds is sharp. Moreover we discuss also the case when $G$ is not connected providing also a sharp lower bound which is linear in both parameter

Ramsey goodness of cycles

Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it is well known and easy to show that $R(G,H) \geq (|G|-1)(\chi(H)-1)+\sigma(H)$, where $\chi(H)$ is the chromatic number of $H$ and $\sigma(H)$ is the size of the smallest color class in a $\chi(H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)= (|G|-1)(\chi(H)-1)+\sigma(H)$. The notion of Ramsey goodness was introduced by Burr and Erd\H{o}s in 1983 and has been extensively studied since then. In this paper we show that if $n\geq 10^{60}|H|$ and $\sigma(H)\geq \chi(H)^{22}$ then the $n$-vertex cycle $C_n$ is $H$-good. For graphs $H$ with high $\chi(H)$ and $\sigma(H)$, this proves in a strong form a conjecture of Allen, Brightwell, and Skokan

On star-wheel Ramsey numbers

For two given graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\bar{G}$ contains a $G_2$. In this note, we determined the Ramsey number $R(K_{1,n},W_m)$ for even $m$ with $n+2\leq m\leq 2n-2$, where $W_m$ is the wheel on $m+1$ vertices, i.e., the graph obtained from a cycle $C_m$ by adding a vertex $v$ adjacent to all vertices of the $C_m$.Comment: 8 page

The Ramsey number of loose cycles versus cliques

Recently Kostochka, Mubayi and Verstra\"ete initiated the study of the Ramsey numbers of uniform loose cycles versus cliques. In particular they proved that $R(C^r_3,K^r_n) = \tilde{\theta}(n^{3/2})$ for all fixed $r\geq 3$. For the case of loose cycles of length five they proved that $R(C_5^r,K_n^r)=\Omega((n/\log n)^{5/4})$ and conjectured that $R(C^r_5,K_n^r) = O(n^{5/4})$ for all fixed $r\geq 3$. Our main result is that $R(C_5^3,K_n^3) = O(n^{4/3})$ and more generally for any fixed $l\geq 3$ that $R(C_l^3,K_n^3) = O(n^{1 + 1/\lfloor(l+1)/2 \rfloor})$. We also explain why for every fixed $l\geq 5$, $r\geq 4$, $R(C^r_l,K^r_n) = O(n^{1+1/\lfloor l/2 \rfloor})$ if $l$ is odd, which improves upon the result of Collier-Cartaino, Graber and Jiang who proved that for every fixed $r\geq 3$, $l\geq 4$, we have $R(C_l^r,K_n^r) = O(n^{1 + 1/(\lfloor l/2 \rfloor-1)})$.Comment: 18 page