On the local density problem for graphs of given odd-girth

TL;DR

This paper confirms Erdős's conjecture on local density for certain triangle-free graphs, extending results to graphs with higher odd-girth and providing degree conditions under which the conjecture holds.

Contribution

It proves Erdős's local density conjecture for graphs homomorphic to Andr ásfai graphs and extends the result to graphs with higher odd-girth, with specific degree conditions.

Findings

Erdős's conjecture holds for graphs with minimum degree > 10n/29.

The conjecture is valid for graphs with chromatic number ≤ 3 and degree > n/3.

Results extend to graphs of higher odd-girth.

Abstract

Erd\H{o}s conjectured that every $n$-vertex triangle-free graph contains a subset of $\lfloor n/2\rfloor$ vertices that spans at most $n^2/50$ edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs homomorphic to so-called Andr\'asfai graphs. As a consequence, Erd\H{o}s' conjecture holds for every triangle-free graph $G$ with minimum degree $\delta (G)>10n/29$ and if $\chi (G)\leq 3$ the degree condition can be relaxed to $\delta (G)> n/3$. In fact, we obtain a more general result for graphs of higher odd-girth.