Extremal bound for 2-resilient 3-graphs

Establish that every 2-resilient 3-uniform hypergraph T with matching number ν(T) = s satisfies |T| ≤ \binom{3s+3}{3} for every positive integer s.

Background

A k-uniform hypergraph is called t-resilient when deleting any set of at most t vertices does not reduce its matching number. The paper focuses on (k−1)-resilient k-graphs and defines m(k,s) as the maximum number of edges in such a hypergraph with matching number s.

For k = 3, the natural complete 3-graph on 3s+2 vertices has \binom{3s+3}{3} edges in the notation displayed in the paper’s conjecture. The authors prove the conjectured bound for s = 2, while the general statement remains unresolved.