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

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.