Tur\'an's problem and generalized Ramsey numbers

TL;DR

This paper investigates a specific class of generalized Ramsey numbers, providing exact values for certain parameters and proposing conjectures to guide future research in graph theory.

Contribution

The paper completely determines the value of $R(n,n(n-1)/2 - r; k, 1)$ for specified parameters and introduces several new conjectures on generalized Ramsey numbers.

Findings

Exact values of $R(n,n(n-1)/2 - r; k, 1)$ for $n \\ge 4$ and $r \\le n-2$

New conjectures on the behavior of generalized Ramsey numbers

Advances understanding of Turán's problem in the context of Ramsey theory

Abstract

Let $n,r,k,s$ be positive integers with $n,k\ge 2$. The generalized Ramsey number $R(n,r;k,s)$ is the smallest positive integer $p$ such that for every graph $G$ of order $p$, either $G$ contains a subgraph induced by $n$ vertices with at most $r-1$ edges, or the complement $\bar G$ of $G$ contains a subgraph induced by $k$ vertices with at most $s-1$ edges. In this paper we completely determine $R(n,n(n-1)/2-r;k,1)$ for $n\ge 4$ and $r\le n-2$, and pose several conjectures on Ramsey numbers.