A hypergraph regularity method for generalised Turan problems

TL;DR

This paper introduces a quasirandom hypergraph counting method for generalized Turan problems, successfully proving a conjecture about the Fano plane's codegree threshold and exploring thresholds for projective planes.

Contribution

It develops a foundational quasirandom counting lemma for hypergraphs and applies it to solve a longstanding conjecture in Turan theory.

Findings

Proved Mubayi's conjecture on the Fano plane codegree threshold.

Established bounds for the codegree thresholds of projective planes.

Identified surprising thresholds for PG_2(4).

Abstract

We describe a method that we believe may be foundational for a comprehensive theory of generalised Turan problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that any 3-graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a Fano plane, for n sufficiently large. For projective planes over fields of odd size q we show that the codegree threshold is between n/2-q+1 and n/2, but for PG_2(4) we find the somewhat surprising phenomenon that the threshold is less than (1/2-c)n for some small c>0. We conclude by setting out a program for future developments of this method to tackle other problems.