On the co-degree threshold for the Fano plane
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.
Cite
@article{arxiv.1204.1927,
title = {On the co-degree threshold for the Fano plane},
author = {Louis DeBiasio and Tao Jiang},
journal= {arXiv preprint arXiv:1204.1927},
year = {2013}
}
Comments
12 pages, 2 figures. This version of the paper contains some additional results obtained using Razborov's flag algebra calculus. We did not include these results in the "European Journal of Combinatorics" version