Mubayi–Pikhurko weakly triangle-free hypergraph extremal problem

Determine whether, for every integer r ≥ 2 and every n, the maximum size of a weakly triangle-free r-uniform hypergraph on n vertices is attained by the balanced complete r-partite r-uniform hypergraph T^r(n).

Background

A weakly triangle-free r-graph is defined as an r-uniform hypergraph containing no three edges A, B, and C such that C contains strictly more than half of the vertices in the symmetric difference A △ B. Mubayi and Pikhurko posed the question of whether the balanced complete r-partite construction T^r(n) is extremal for this class.

The paper proves a stronger result for the related family Δ_r: for each fixed r and all sufficiently large n, T^r(n) is the unique extremal Δ_r-free construction. Since weakly triangle-free r-graphs are Δ_r-free, this resolves the Mubayi–Pikhurko question in the large-n regime, but the question as stated for arbitrary n is not established by the paper.