Complementary Ramsey numbers and Ramsey graphs

TL;DR

This paper introduces a new variant of Ramsey numbers called complementary Ramsey numbers, explores their properties, and determines specific values for small parameters by classifying related Ramsey graphs.

Contribution

It defines complementary Ramsey numbers and computes their values for certain small cases using classification of Ramsey graphs.

Findings

Determined $ar{R}(m,4,4)$ and $ar{R}(m,3,6)$ for small parameters.

Established connections between complementary Ramsey numbers and pairs of Ramsey graphs.

Provided classifications of Ramsey (s,t)-graphs for small s,t.

Abstract

In this paper, we consider a variant of Ramsey numbers which we call complementary Ramsey numbers $\bar{R}(m,t,s)$. We first establish their connections to pairs of Ramsey $(s,t)$-graphs. Using the classification of Ramsey $(s,t)$-graphs for small $s,t$, we determine the complementary Ramsey numbers $\bar{R}(m,t,s)$ for $(s,t)=(4,4)$ and $(3,6)$.