Erdős Matching Conjecture for uniform set families

Prove the Erdős Matching Conjecture in full generality: determine whether every family mathcal{F}\subseteq\binom{[n]}{k} of size greater than the conjectured extremal bound contains an s-matching, for all admissible values of n, k, and s.

Background

The paper places rainbow matching questions in the context of the classical Erdős Matching Conjecture, which concerns the maximum size of a k-uniform family on [n] that contains no s pairwise disjoint sets. The conjecture remains unresolved in general, although several partial results are known. The paper contrasts this unresolved non-partite problem with its k-partite analogue, whose corresponding one-family statement is easy to prove by averaging.