Exact optimality of the quasi-bipartite construction among cross-edge-free hypergraphs

Determine whether the quasi-bipartite hypergraph H_{r,d} is exactly optimal for the number of weak independent sets among all r-uniform, d-regular hypergraphs with no cross-edges, including the case r=3.

Background

The paper studies the Balogh–Bollobás–Narayanan conjecture that H_{r,d} maximizes the number of weak independent sets among all finite simple r-uniform, d-regular hypergraphs. It proves asymptotic upper bounds under pair-codegree or cross-edge-free hypotheses and establishes exact results for d=2 when r is odd and for (r,d)=(4,2).

The concluding remark identifies a remaining exact extremal question even within the more restricted class of cross-edge-free hypergraphs. In particular, the authors explicitly state that exact optimality is unresolved already for 3-uniform hypergraphs.