Ramsey lower bounds for bounded degree hypergraphs

TL;DR

This paper establishes new lower bounds for the Ramsey numbers of bounded degree hypergraphs, showing they grow at least as fast as a tower function, advancing a longstanding open problem.

Contribution

It provides the first construction demonstrating that Ramsey numbers for bounded degree hypergraphs grow at least as fast as a tower function, addressing a question from 2009.

Findings

Proves existence of hypergraphs with large Ramsey numbers and bounded degree.

Shows lower bounds grow as a tower function of degree and maximum degree.

Advances understanding of hypergraph Ramsey theory.

Abstract

We prove that for all $k \ge 3$ and any integers $\Delta, n$ with $n \ge 2^\Delta,$ there exists a $k$-graph on $n$ vertices with maximum degree at most $\Delta$ such that $r(H)\geq\tw_{k-1}(c_k \Delta) \cdot n$ for some constant $c_k > 0$, where $\tw_k$ denotes the tower function. This makes the first progress toward a problem proposed by Conlon, Fox, and Sudakov (2009), who asked whether $r(H)\geq\tw_{k}(c_k \Delta) \cdot n$ holds. Our proof relies on a novel construction of a $k$-graph on a growing number of vertices $n$ while keeping the maximum degree bounded by a fixed $\Delta$.