Improved bounds on the multicolor Ramsey numbers of paths and even cycles

TL;DR

This paper improves the upper bounds on multicolor Ramsey numbers for paths and even cycles, providing tighter estimates and new structural insights into connected graphs without large matchings.

Contribution

The authors establish a sharper upper bound of rac{1}{2} + o(1) for the multicolor Ramsey numbers, advancing previous bounds and introducing structural analysis techniques.

Findings

Improved upper bound of (k - 1/2 + o(1))n for R_k(P_n) and R_k(C_n)

Structural insights into connected graphs without large matchings

Potential applicability of methods to related combinatorial problems

Abstract

We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. For a long time, $R_k(P_n)$ has only been known to lie between $(k-1+o(1))n$ and $(k + o(1))n$. A recent breakthrough by S\'ark\"ozy and later improvement by Davies, Jenssen and Roberts give an upper bound of $(k - \frac{1}{4} + o(1))n$. We improve the upper bound to $(k - \frac{1}{2}+ o(1))n$. Our approach uses structural insights in connected graphs without a large matching. These insights may be of independent interest.