Multipartite Ramsey numbers of complete bipartite graphs arising from algebraic combinatorial structures

TL;DR

This paper extends the understanding of multipartite Ramsey numbers for complete bipartite graphs by introducing a new class of matrices and establishing results that generalize previous findings, addressing open problems in the field.

Contribution

It introduces the concept of $[eta]$-Hadamard matrices, conjectures their existence for all orders, and determines new set and size multipartite Ramsey numbers for specific graphs.

Findings

Generalized previous results using $[eta]$-Hadamard matrices

Conjectured existence of $[eta]$-Hadamard matrices for all orders

Determined new multipartite Ramsey numbers for certain graphs

Abstract

In 2019, Perondi and Carmelo determined the set multipartite Ramsey number of particular complete bipartite graphs by establishing a relationship between the set multipartite Ramsey number, Hadamard matrices, and strongly regular graphs, which is a breakthrough in Ramsey theory. However, since Hadamard matrices of order not divisible by 4 do not exist, many open problems have arisen. In this paper, we generalize Perondi and Carmelo's results by introducing the $[\alpha]$-Hadamard matrix that we conjecture exists for arbitrary order. Finally, we determine set and size multipartite Ramsey numbers for particular complete bipartite graphs.