Ramsey numbers of sparse hypergraphs

TL;DR

This paper presents a simplified, self-contained proof that the Ramsey number of bounded-degree k-uniform hypergraphs grows linearly with the number of vertices, improving previous bounds and providing new techniques.

Contribution

The authors introduce a new, simpler proof method for bounding hypergraph Ramsey numbers, replacing complex regularity-based proofs with innovative techniques and sharper bounds.

Findings

Bounded-degree hypergraphs have linear Ramsey numbers in the number of vertices.

The new bounds for k ≥ 4 are tower-type, significantly better than previous Ackermann-type bounds.

The methods yield sharp results for hypergraphs with a limited number of edges.

Abstract

We give a short proof that any k-uniform hypergraph H on n vertices with bounded degree \Delta has Ramsey number at most c(\Delta, k)n, for an appropriate constant c(\Delta, k). This result was recently proved by several authors, but those proofs are all based on applications of the hypergraph regularity method. Here we give a much simpler, self-contained proof which uses new techniques developed recently by the authors together with an argument of Kostochka and R\"odl. Moreover, our method demonstrates that, for k \geq 4, c(\Delta, k) \leq 2^{2^{\Ddots^{2^{c \Delta}}}}, where the tower is of height k and the constant c depends on k. It significantly improves on the Ackermann-type upper bound that arises from the regularity proofs, and we present a construction which shows that, at least in certain cases, this bound is not far from best possible. Our methods also allows us to prove quite sharp results on the Ramsey number of hypergraphs with at most m edges.