3-Uniform hypergraphs of bounded degree have linear Ramsey numbers

TL;DR

This paper extends the linear bound on Ramsey numbers from graphs to 3-uniform hypergraphs with bounded degree, using a new embedding lemma for pseudo-random hypergraphs.

Contribution

It introduces a novel embedding lemma for 3-uniform hypergraphs of bounded degree into pseudo-random hypergraphs, establishing linear Ramsey numbers for these hypergraphs.

Findings

Ramsey numbers of 3-uniform hypergraphs with bounded degree are linear in their size

Developed an embedding lemma for hypergraphs into pseudo-random structures

Extended known graph results to hypergraph setting

Abstract

Chv\'atal, R\"odl, Szemer\'edi and Trotter proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. We prove that the same holds for 3-uniform hypergraphs. The main new tool which we prove and use is an embedding lemma for 3-uniform hypergraphs of bounded maximum degree into suitable 3-uniform `pseudo-random' hypergraphs.