On Some Three-Color Ramsey Numbers for Paths

TL;DR

This paper investigates three-color Ramsey numbers for paths and cycles, proving new exact values and exploring conjectures that relate these numbers, thereby advancing understanding in combinatorial graph theory.

Contribution

The paper proves new exact values for three-color Ramsey numbers for paths without computer assistance and links conjectured equalities for cycles to those for paths.

Findings

Established $R(P_8,P_8,P_8)=14$ and $R(P_9,P_9,P_9)=17$

Connected conjectured cycle Ramsey numbers to path Ramsey numbers

Supported the conjecture that $R(P_n,P_n,P_n)=2n-2+(n\bmod 2)$ for all $n\ge1$

Abstract

For graphs $G_1, G_2, G_3$, the three-color Ramsey number $R(G_1,$ $G_2, G_3)$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with 3 colors, then it contains a monochromatic copy of $G_i$ in color $i$, for some $1 \leq i \leq 3$. First, we prove that the conjectured equality $R_3(C_{2n},C_{2n},C_{2n})=4n$, if true, implies that $R_3(P_{2n+1},P_{2n+1},P_{2n+1})=4n+1$ for all $n \ge 3$. We also obtain two new exact values $R(P_8,P_8,P_8)=14$ and $R(P_9,P_9,P_9)=17$, furthermore we do so without help of computer algorithms. Our results agree with a formula $R(P_n,P_n,P_n)=2n-2+(n\bmod 2)$ which was proved for sufficiently large $n$ by Gy\'arf\'as, Ruszink\'o, S\'ark\"ozy, and Szemer\'{e}di in 2007. This provides more evidence for the conjecture that the latter holds for all $n \ge 1$.