Frankl–Huang–Rödl conjecture for the pair of forbidden 3-graphs

Determine whether there exists an integer m_0 such that, for every integer m>m_0, the extremal number of the 3-graph family {R_1,R_2}, where R_1=F_{4,3}\setminus\{156,257,367\} and R_2=F_{4,3}\setminus\{156,157,367\}, satisfies ex(2m,{R_1,R_2})=\binom{2m}{3}-2\binom{m}{3}.

Background

The paper defines F_{4,3} as the 3-uniform hypergraph consisting of a four-vertex core, a three-vertex outer set, all four triples within the core, and all twelve triples meeting the core in one vertex and the outer set in two vertices. The two hypergraphs R_1 and R_2 are obtained by deleting three specified triples from F_{4,3}. Both are subgraphs of F^-_{4,3}, the hypergraph obtained by deleting one such triple from F_{4,3}, and neither R_1 nor R_2 is two-colorable.

The balanced complete bipartite 3-graph B_n is free of both R_1 and R_2 and has b(n)=\binom n3-\binom{\lfloor n/2\rfloor}{3}-\binom{\lceil n/2\rceil}{3} edges. Frankl, Huang, and Rödl conjectured that for sufficiently large even order 2m, this construction is extremal. The paper proves a stronger result for every n\ge 8: ex(n,{R_1,R_2})=b(n), with B_n as the unique extremal 3-graph, thereby proving the conjecture with m_0=3.