Pentagons in triangle-free graphs

TL;DR

This paper proves that for all sufficiently large triangle-free graphs with at least 9 vertices, the maximum number of 5-cycles is achieved by balanced blow-ups of a 5-cycle, confirming a longstanding conjecture.

Contribution

It fully resolves Erd ext{"o}s's conjecture by characterizing extremal graphs for all large enough n, extending previous partial results.

Findings

Balanced blow-ups of a 5-cycle maximize 5-cycles in triangle-free graphs.

The result holds for all n ≥ 9, not just large or divisible by 5.

Complete resolution of a conjecture by Erd ext{"o}s.

Abstract

For all $n\ge 9$, we show that the only triangle-free graphs on $n$ vertices maximizing the number $5$-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by Erd\H{o}s, and extends results by Grzesik and Hatami, Hladk\'y, Kr\'{a}l', Norin and Razborov, where they independently showed this same result for large $n$ and for all $n$ divisible by $5$.