On the Ramsey number of daisies II

TL;DR

This paper establishes a nearly optimal exponential lower bound for the Ramsey number of a specific class of hypergraphs called daisies, extending previous results and matching bounds known for complete hypergraphs.

Contribution

It provides an $(r-2)$-iterated exponential lower bound for the Ramsey number of $(r,m,k)$-daisies in 2-colorings, advancing understanding of hypergraph Ramsey theory.

Findings

Lower bound matches the order of magnitude of complete hypergraph Ramsey numbers

Extends previous work on daisies with improved bounds

Uses combinatorial and probabilistic methods to establish bounds

Abstract

A $(k+r)$-uniform hypergraph $H$ on $(k+m)$ vertices is an $(r,m,k)$-daisy if there exists a partition of the vertices $V(H)=K\cup M$ with $|K|=k$, $|M|=m$ such that the set of edges of $H$ is all the $(k+r)$-tuples $K\cup P$, where $P$ is an $r$-tuple of $M$. Complementing results in ["On the Ramsey number of daisies I"], we obtain an $(r-2)$-iterated exponential lower bound to the Ramsey number of an $(r,m,k)$-daisy for $2$-colors. This matches the order of magnitude of the best lower bounds for the Ramsey number of a complete $r$-graph.