On the co-degree threshold for the Fano plane

Given a 3-graph H, let ex2(n,H) denote the maximum value of the minimum co-degree 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 (2005)  [14] proved that ex2(n,F)=(1/2+o(1))n and conjectured that ex2(n,F)=⌊n/2⌋ for sufficiently large n. Using a very sophisticated quasi-randomness argument, Keevash (2009)  [7] 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 Turán density of the family of rings.