Bounds for Gallai-Ramsey functions and numbers

TL;DR

This paper establishes new upper bounds for Gallai-Ramsey functions and numbers, especially for complete graphs, and provides lower bounds using probabilistic methods, advancing understanding of edge-coloring Ramsey problems.

Contribution

It improves upper bounds for Gallai-Ramsey numbers for complete graphs and extends bounds to special graph cases, also deriving lower bounds via Lovász Local Lemma.

Findings

Proved $ ext{gr}_k(K_s,K_t) ext{ } extless{} 2^{kt}s^{3kt}$ for $t extgreater{} 47$.

Provided better bounds for specific graph pairs.

Derived lower bounds using probabilistic techniques.

Abstract

For two graphs $G,H$ and a positive integer $k$, the \emph{Gallai-Ramsey number} $\operatorname{gr}_k(G,H)$ is defined as the minimum number of vertices $n$ such that any $k$-edge-coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a monochromatic copy of $H$. If $G$ and $H$ are both complete graphs, then we call it Gallai-Ramsey function. Fox and Sudakov proved $\operatorname{gr}_k(K_s,K_t)\leq s^{4kt}$. Alon et al. showed that $\operatorname{gr}_k(K_s,K_t)\leq (2s^3+4s^2)^{kt}$. In this paper, we prove that $\operatorname{gr}_k(K_s,K_t)\leq 2^{kt}s^{3kt}$ for $t\geq 47$. We also give better upper bounds for $\operatorname{gr}_k(G,H)$ when $G,H$ are some special graphs. In this paper, we derive some lower bounds for Gallai-Ramsey functions and numbers by Lov\'{a}sz Local Lemma.