Projective planes and set multipartite Ramsey numbers for $C_4$ versus star

TL;DR

This paper investigates set multipartite Ramsey numbers involving a 4-cycle and a star, establishing bounds, exploring their relationship with classical Ramsey numbers, and utilizing projective plane polarity graphs to find optimal classes.

Contribution

It introduces bounds and exact classes for $C_4$-star set multipartite Ramsey numbers and employs projective plane polarity graphs to identify optimal configurations.

Findings

Derived near-optimal and exact classes of set multipartite Ramsey numbers.

Established relationships between set multipartite and classical Ramsey numbers.

Utilized polarity graphs from projective planes to find optimal subgraphs.

Abstract

Set multipartite Ramsey numbers were introduced in 2004, ge-neralizing the celebrated Ramsey numbers. Let $C_4$ denote the four cycle and let $K_{1,n}$ denote the star on $n+1$ vertices. In this paper we investigate bounds on $C_4-K_{1,n}$ set multipartite Ramsey numbers. Relationships between these numbers and the classical $C_4-K_{1,n}$ Ramsey numbers are explored. Then several near-optimal or exact classes are derived as applications. As the main goal, polarity graphs from projective planes allow us to find suitable subgraphs which yield some optimal classes too.