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

TL;DR

This paper determines exact multicolor Ramsey numbers for paths, cycles, stripes, and stars using bipartite Ramsey numbers, addressing many open cases and providing new results for specific graph configurations.

Contribution

The paper derives exact multicolor Ramsey numbers for various graph classes, expanding knowledge on these numbers and employing bipartite Ramsey numbers as a key tool.

Findings

Exact values for R(C_{n_0}, P_{n_1}, P_{n_2}) under certain conditions

R(P_n, kK_2, kK_2) = n + 2k - 2 for large n

R((k-1)K_2, P_k, P_k) = 3k - 4 for even k

Abstract

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.