On the Ramsey number of daisies I

TL;DR

This paper investigates the Ramsey numbers of daisies, a special class of hypergraphs, providing bounds and insights into their combinatorial properties, extending classical Ramsey theory to these structures.

Contribution

It introduces bounds for the Ramsey numbers of daisies, a new class of hypergraphs, and explores cases with bounded kernel sizes, advancing understanding of their combinatorial complexity.

Findings

Established upper bounds for daisy Ramsey numbers.

Derived lower bounds for specific daisy configurations.

Extended classical Ramsey theory to hypergraph structures called daisies.

Abstract

Daisies are a special type of hypergraphs introduced by Bollob\'{a}s, Leader and Malvenuto. An $r$-daisy determined by a pair of disjoint sets $K$ and $M$ is the $(r+|K|)$-uniform hypergraph $\{K\cup P:\: P\in M^{(r)}\}$. In [Combin. Probab. Comput. 20, no. 5, 743-747, 2011] the authors studied Tur\'{a}n type density problems for daisies. This paper deals with Ramsey numbers of Daisies, which are natural generalizations of classical Ramsey numbers. We discuss upper and lower bounds for the Ramsey number of $r$-daisies and also for special cases where the size of the kernel is bounded.