Bound for critically intersecting hypergraphs

Establish whether the bound |F| ≤ \binom{2k−t}{k} holds for every t-intersecting k-uniform hypergraph F with the stated covering property whenever k > c(k−t) for some absolute constant c.

Background

The paper recalls an analogous problem for t-intersecting k-graphs with covering number k. It states a known theorem proving the bound under the stronger condition k ≥ (k−t)^4 and characterizing equality.

The authors then record the unresolved conjectural strengthening that the same conclusion should hold under a linear threshold k > c(k−t), where c is an absolute constant.