Erdős Matching Conjecture

Prove the Erdős Matching Conjecture: for every positive integer s, every integer n ≥ (s + 1)k, and every k-uniform hypergraph F on [n] with matching number ν(F) ≤ s, establish that |F| ≤ max{\binom{n}{k} − \binom{n−s}{k}, \binom{(s+1)k−1}{k}}.

Background

The paper places resilient hypergraph questions in the context of the Erdős Matching Conjecture, which seeks the largest possible size of a k-uniform hypergraph with matching number at most s. The conjectured extremal bound is given by the larger of two natural constructions: all k-sets intersecting a fixed s-set, and all k-sets contained in a set of size (s+1)k−1.

The paper notes that the conjecture has been proved in several important regimes, including the case k = 3, but presents it as a major open problem in general. This provides the broader extremal-set-theoretic context for the authors’ study of resilient hypergraphs.