On the co-degree threshold for the Fano plane

Given a 3-graph H , let ex 2 ( 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 { a x x ′ , a y y ′ , a z z ′ , x y z ′ , x y ′ z , x ′ y z , x ′ y ′ z ′ } . Mubayi (2005)  [14] proved that ex 2 ( n , F ) = ( 1 / 2 + o ( 1 ) ) n and conjectured that ex 2 ( 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.