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

For given graphs $G_1,G_2,\mathrm{.}\mathrm{.}\mathrm{.},G_k,k\ge 2$, the multicolor Ramsey number $R(G_1,G_2,\mathrm{.}\mathrm{.}\mathrm{.},G_k)$ 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 $1\le i\le k$. The bipartite Ramsey number $b(G_1, \cdots, G_k)$istheleastpositiveinteger$b$suchthatanycoloringoftheedgesof$K_{b,b}$with$k$colorswillresultinamonochromaticcopyofbipartite$G_i$inthe$i$-thcolor,forsome$i$,$1 \le i \le k$. There is very little known about $R(G_1,\dots ,G_k)$ 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 $$R(C_{n_0},P_{n_1},P_{n_2})=n_0+\lfloor \frac{n_1}{2}\rfloor +\lfloor \frac{n_2}{2}\rfloor -2.$$ We prove that $R(P_n,k{K}_2,k{K}_2)=n+2k-2$ 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 $t\ge m+s-1$, we obtain that $R(t{K}_2,P_{2s+1},P_{2m})=s+m+2t-2$ where $P_k$ is a path on $k$ vertices and $t{K}_2$ 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.