Constructive Methods in Gallai-Ramsey Theory for Hypergraphs

TL;DR

This paper explores constructive methods in hypergraph Gallai-Ramsey theory, providing new lower bounds for specific hypergraph Ramsey numbers through innovative constructions and focusing on colorings avoiding rainbow subhypergraphs.

Contribution

It introduces new constructions that improve lower bounds for 3- and 4-uniform hypergraph Ramsey and Gallai-Ramsey numbers, advancing the understanding of hypergraph coloring constraints.

Findings

New lower bounds for 3- and 4-uniform Ramsey numbers

Improved bounds for Gallai-Ramsey numbers in hypergraphs

Constructive methods for hypergraph coloring avoiding rainbow subhypergraphs

Abstract

Much recent progress in hypergraph Ramsey theory has focused on constructions that lead to lower bounds for the corresponding Ramsey numbers. In this paper, we consider applications of these results to Gallai colorings. That is, we focus on the Ramsey numbers resulting from only considering $t$-colorings of the hyperedges of complete $r$-uniform hypergraphs in which no rainbow $K_{r+1}^{(r)}$-subhypergraphs exists. We also provide new constructions which imply improved lower bounds for many $3$ and $4$-uniform Ramsey numbers and $3$ and $4$-uniform Gallai-Ramsey numbers.