# Diagonal Ramsey numbers in multipartite graphs related to stars

## TL;DR

This paper determines the exact size Ramsey numbers for multipartite graphs with respect to stars, providing key insights into edge colorings that force the presence of star subgraphs.

## Contribution

It establishes the exact values of size Ramsey numbers for complete balanced multipartite graphs concerning stars, a novel result in this area.

## Key findings

- Exact values of $m_j(S_n,S_m)$ for $n,m
geq 3$ and $j
geq 3$
- Characterization of edge colorings avoiding certain star subgraphs
- Advancement in understanding Ramsey properties of multipartite graphs

## Abstract

Let the star on $n$ vertices, namely $K_{1,n-1}$ be denoted by $S_n$. If every two coloring of the edges of a complete balanced multipartite graph $K_{j \times s}$ there is a copy of $S_n$ in the first color or a copy of $S_m$ in the second color, then we will say $K_{j \times s}\rightarrow (S_n,S_m)$. The size Ramsey multipartite number $m_j(S_n, S_m)$ is the smallest natural number $s$ such that $K_{j \times s}\rightarrow (S_n,S_m)$. In this paper, we obtain the exact values of the size Ramsey numbers $m_j(S_n,S_m)$ for $n,m \ge 3$ and $j \ge 3$.