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.

References

Erd\H{o}s posed a well-known conjecture, commonly known as the Erd\H{o}s Matching Conjecture, concerning the exact value of m(n,k,t). This conjecture remains open in general.

— Extremal problems for cancellative and locally thin hypergraphs  (2609.08858 - Liu et al., 8 Sep 2026) in Section 2, subsection “Cancellative \(r\)-graphs,” Section 5, subsection “The upper bound”

One of the classical questions in extremal set theory is the Erdős Matching Conjecture [6]. It suggests the size of the largest family in \binom{[n]}{k} that contains no s pairwise disjoint sets (an s-matching). In spite of the efforts by different researchers, it is still open in general.

— Satisfying sequences for rainbow partite matchings  (2502.03105 - Kupavskii et al., 5 Feb 2025) in Section 1, page 2