Erdős Matching Conjecture

The Erdős matching conjecture asserts that an -uniform hypergraph on vertices with no pairwise vertex-disjoint hyperedges has at most

hyperedges, for integers , , and (Erdős 1965, Frankl 2017).

Both candidate bounds are attained. One construction takes all -subsets meeting a fixed -element set; the other takes all -subsets of a set of vertices, leaving the remaining vertices isolated. The conjecture remains open in general (Bloom 2026).

This is problem 1020 in the Erdős problems collection (Bloom 2026).