Large-covering-number bound for resilient k-graphs

Prove that every k-uniform hypergraph F with matching number ν(F) = s and covering number τ(F) = sk satisfies |F| ≤ \binom{sk+k−1}{k} for all sufficiently large s depending on k.

Background

Conjecture 7.2 is presented as a weaker version of the preceding asymptotic conjecture. The condition τ(F) = sk is the covering-number formulation naturally associated with (k−1)-resilience in the intersecting case and specifies the class of hypergraphs under consideration.

The proposed bound is the size of the complete k-graph on sk+k−1 vertices, so the conjecture asks whether this natural construction is asymptotically extremal under the stated matching and covering-number conditions.