Counting flags in triangle-free digraphs

TL;DR

This paper proves that digraphs with a minimum outdegree of at least approximately 0.3465n necessarily contain an oriented triangle, advancing previous bounds and utilizing flag algebra theory.

Contribution

It improves the minimum outdegree bound for guaranteeing an oriented triangle in triangle-free digraphs using flag algebra methods.

Findings

Minimum outdegree threshold for oriented triangles is approximately 0.3465n.

Improves previous bound of 0.3532n.

Utilizes flag algebra theory in the proof.

Abstract

Motivated by the Caccetta-Haggkvist Conjecture, we prove that every digraph on n vertices with minimum outdegree 0.3465n contains an oriented triangle. This improves the bound of 0.3532n of Hamburger, Haxell and Kostochka. The main new tool we use in our proof is the theory of flag algebras developed recently by Razborov.