The next case of Andr\'asfai's conjecture

TL;DR

This paper advances the understanding of Andre1sfasai's conjecture by determining the maximum edges in certain triangle-free graphs with bounded independence number for a new range of parameters.

Contribution

It proves a new explicit formula for ex(n,s) for s/n in [4/11, 3/8], extending previous results and moving closer to settling Andre1sfasai's conjecture.

Findings

Established ex(n,s)=6n^2-32ns+44s^2 for s/n in [4/11, 3/8]

Extended the known ranges where ex(n,s) is explicitly determined

Progressed towards confirming Andre1sfasai's conjecture for triangle-free graphs

Abstract

Let $\mathrm{ex}(n,s)$ denote the maximum number of edges in a triangle-free graph on $n$ vertices which contains no independent sets larger than $s$. The behaviour of $\mathrm{ex}(n,s)$ was first studied by Andr\'asfai, who conjectured that for $s>n/3$ this function is determined by appropriately chosen blow-ups of so called Andr\'asfai graphs. Moreover, he proved $\mathrm{ex}(n, s)=n^2-4ns+5s^2$ for $s/n\in [2/5, 1/2]$ and in earlier work we obtained $\mathrm{ex}(n, s)=3n^2-15ns+20s^2$ for $s/n\in [3/8, 2/5]$. Here we make the next step in the quest to settle Andr\'asfai's conjecture by proving $\mathrm{ex}(n, s)=6n^2-32ns+44s^2$ for $s/n\in [4/11, 3/8]$.