Minimum degrees and codegrees of minimal Ramsey 3-uniform hypergraphs

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, Erdős and Lovasz [S. A. Burr, P. Erdős, and L. Lovász, On graphs of Ramsey type, Ars Combinatoria 1 (1976), no. 1, 167–190] 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 Kt(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.