On three-color Ramsey number of paths

TL;DR

This paper determines the exact three-color Ramsey numbers for paths, extending known results and providing formulas for specific cases involving paths and matchings.

Contribution

It establishes new exact values for multicolor Ramsey numbers involving paths, including cases with three colors and specific path and matching configurations.

Findings

Proves that $R(P_3, P_n, P_m) = R(P_n, P_m) = m + loor{rac{n}{2}} - 1$ for most cases.

Derives that $R(P_3, mK_2, nK_2) = 2m + n - 1$ for $m \\geq n \\geq 3$.

Abstract

Let $G_1, G_2, ..., G_t$ be graphs. The multicolor Ramsey number $R(G_1, G_2, ..., G_t)$ is the smallest positive integer $n$ such that if the edges of complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs $H_1,H_2,...,H_t$, then at least one $H_i$ has a subgraph isomorphic to $G_i$. In this paper, we prove that if $(n,m)\neq (3,3), (3,4)$ and $m\geq n$, then $R(P_3,P_n,P_m)=R(P_n,P_m)=m+\lfloor \frac{n}{2}\rfloor-1$. Consequently $R(P_3,mK_2,nK_2)=2m+n-1$ for $m\geq n\geq 3$.