Multicolor, multipartite Ramsey numbers for quadrilateral

Janusz Dybizbański; Yaser Rowshan

paper

Multicolor, multipartite Ramsey numbers for quadrilateral

arXiv:2402.16816

Abstract

The $p$ -partite Ramsey number for quadrilateral, denoted by $r_{p} (C_{4}, k)$ , is the least positive integer $n$ such that any coloring of the edges of a complete $p$ -partite graph with $n$ vertices in each partition with $k$ colors will result in a monochromatic copy of $C_{4}$ . In this paper, we present an upper bound for $r_{p} (C_{4}, k)$ and the exact values of $r_{p} (C_{4}, 2)$ for all $p \geq 2$ . In tripartite case we show that $r_{3} (C_{4}, k) \leq ⌊(k + 1)^{2} /2 ⌋ - 1$ and the exact value of 4-color tripartite Ramsey number $r_{3} (C_{4}, 4) = 11$ .