Erdős Matching Conjecture

Erdős posed the conjecture in 1965: it predicts the largest family of $k$-sets containing no $s+1$ pairwise disjoint members, for $n\ge (s+1)k$.

Known results

• Kleitman proved the boundary case $n=sk$.
• The cases $k=2$ and $k=3$ are solved.
• For sufficiently large $s$, the cover-family bound is sharp when $n\ge \frac{5}{3}sk-\frac{2}{3}s$ (2018).
• For $k\ge 5$ and sufficiently large $s$, the first extremal construction is proved near $n=(s+1)k$ (2022).

February 2026 claimed proof; August 2026 partial advance

Mishra's February 1, 2026 preprint claims a complete proof via sequential shifting, but this remains unverified. In August 2026, Cao, Liu, and Zhang proved the conjectured bound for fixed $k$ and sufficiently large $s$ when $n\ge (k+1)s$, with stability; they explicitly leave the full conjecture unresolved outside that range.

Current status (as of August 2026): A complete proof is claimed but unverified; substantial ranges are settled, while the general conjecture remains open outside them.