On Ramsey number of Steiner systems

TL;DR

This paper proves the existence of certain hypergraphs called partial $(k,k-1)$-systems with Ramsey numbers that grow extremely rapidly, as a tower of height $k-1$, for any number of colors $r \,\geq\, 4$.

Contribution

It establishes the existence of partial $(k,k-1)$-systems with super-exponentially large Ramsey numbers, advancing understanding of hypergraph Ramsey theory.

Findings

Ramsey number grows as a tower of height $k-1$

Existence of such hypergraphs for $r \geq 4$ colors

Significant growth rate of hypergraph Ramsey numbers

Abstract

A $k$-uniform hypergraph $H$ is called a partial $(k,\ell)$-system if every set of $\ell$ vertices of $V(H)$ is contained in at most one edge of $H$. We prove the existence of a partial $(k,k-1)$-system $H$ whose Ramsey number with $r \geq 4$ colors grows as a tower of height $k-1$.