On the size multipartite Ramsey numbers involving complete graphs

TL;DR

This paper investigates the size multipartite Ramsey number mj(H, G), providing bounds and exact values for specific graph pairs, advancing understanding of how graph properties influence these numbers.

Contribution

It establishes new bounds and exact values for mj(H, G), especially for complete graphs and graphs with large maximum degree, based on chromatic number and graph order.

Findings

Derived a lower bound for mj(H, G) using chromatic number and order.

Obtained a tight lower bound for mj(Km, G) with large maximum degree.

Determined the order of magnitude and exact values of mj(Km, K1,n) for specific parameters.

Abstract

Given two graphs H and G, the size multipartite Ramsey number mj (H, G) is the smallest natural number t such that an arbitrary coloring of the edges of Kjt, complete multipartite graph whose vertex set is partitioned into j parts each of size t, using two colors red and blue, necessarily forces a red copy of H or a blue copy of G as a subgraph. The notion of size multipartite Ramsey number has been introduced by Burger and Vuuren in 2004. It is worth noting that, this concept is derived by using the idea of the original classical Ramsey number, multipartite Ramsey number and the size Ramsey number. In this paper, we focus on mj(H, G) and find a lower bound for mj (H, G) based on the chromatic number of H and the order of G. Also, for graphs G with large maximum degree, we obtain a tight lower bound for mj(Km, G). Furthermore, we determine the order of magnitude of mj(Km, K1,n), for j >= m >= 3 and n >= 2. Then we specify the exact values of mj(Km, K1,n) for the cases m = j, m = 3 and j = 0 or m-2 (mod m-2).