Ramsey numbers for regular induced subgraphs

TL;DR

This paper investigates the minimum size of graphs that guarantee the presence of regular induced subgraphs of specified orders, providing exact values for small sizes and improved bounds for larger ones.

Contribution

It offers exact values for regular induced subgraphs of sizes up to 5 and enhances existing lower bounds for sizes 6 and 7.

Findings

Exact values for regular induced subgraphs of size up to 5.

Lower bounds established for sizes 6 and 7.

Improved general lower bound for regular induced subgraphs.

Abstract

A problem proposed by Erd\H{o}s, Fajtlowicz and Staton asks for the smallest $n$ for which every graph on $n$ vertices contains a regular induced subgraph of order at least $k$. A variation is to ask for a regular induced subgraph of order exactly $k$. In this paper we provide exact values for $k\le 5$ and lower bounds for $k=6$ and $k=7$. We also improve the general lower bound of Alon, Krivelevich and Sudakov [SIAM J. Disc. Math, 2008].