On the co-degree threshold for the Fano plane

TL;DR

This paper proves Mubayi's conjecture that the maximum minimum codegree in 3-graphs avoiding a Fano plane is roughly half the number of vertices, using a new approach involving rings.

Contribution

The paper provides a simple proof of Mubayi's conjecture on the co-degree threshold for the Fano plane, introducing the concept of rings and determining their Turán density.

Findings

Confirmed Mubayi's conjecture with a simplified proof.

Introduced the class of rings in 3-graphs.

Determined the Turán density of rings.

Abstract

Given a 3-graph H, let \ex_2(n, H) denote the maximum value of the minimum codegree of a 3-graph on n vertices which does not contain a copy of H. Let F denote the Fano plane, which is the 3-graph \{axx',ayy',azz',xyz',xy'z,x'yz,x'y'z'\}. Mubayi proved that \ex_2(n,F)=(1/2+o(1))n and conjectured that \ex_2(n, F)=\floor{n/2} for sufficiently large n. Using a very sophisticated quasi-randomness argument, Keevash proved Mubayi's conjecture. Here we give a simple proof of Mubayi's conjecture by using a class of 3-graphs that we call rings. We also determine the Tur\'an density of the family of rings.