Minimum degrees and codegrees of minimal Ramsey 3-uniform hypergraphs

TL;DR

This paper investigates the minimal degree and codegree parameters of minimal Ramsey 3-uniform hypergraphs, revealing exponential bounds and advancing understanding of their structural properties in hypergraph Ramsey theory.

Contribution

It introduces the study of minimum degrees and codegrees in minimal Ramsey 3-uniform hypergraphs, providing bounds and new insights into their combinatorial structure.

Findings

Smallest minimum vertex degree is exponential in polynomial of k and t.

Analyzes minimal 2-Ramsey 3-uniform hypergraphs for codegree bounds.

Advances understanding of hypergraph Ramsey parameters.

Abstract

A uniform hypergraph $H$ is called $k$-Ramsey for a hypergraph $F$, if no matter how one colors the edges of $H$ with $k$ colors, there is always a monochromatic copy of $F$. We say that $H$ is minimal $k$-Ramsey for $F$, if $H$ is $k$-Ramsey for $F$ but every proper subhypergraph of $H$ is not. Burr, Erd\H{o}s and Lovasz studied various parameters of minimal Ramsey graphs. In this paper we initiate the study of minimum degrees and codegrees of minimal Ramsey $3$-uniform hypergraphs. We show that the smallest minimum vertex degree over all minimal $k$-Ramsey $3$-uniform hypergraphs for $K_t^{(3)}$ is exponential in some polynomial in $k$ and $t$. We also study the smallest possible minimum codegrees over minimal $2$-Ramsey $3$-uniform hypergraphs.