Erdős Matching Conjecture

The conjecture, posed by Erdős in 1965, predicts the exact largest size of a family of $k$-sets containing no $s$ pairwise disjoint members. The conjectured bound is known in several cases, but a general proof has not been independently confirmed.

Known results

• Kleitman proved the boundary case $n=sk$.
• The cases $k=2$ and $k=3$ are solved.
• Earlier work established sufficiently-large-$n$ ranges, including results of Erdős, Bollobás–Daykin–Erdős, and Huang–Lo–Sudakov.
• Stability results cover ranges including $n\geq (2+o(1))sk$ as $s\to\infty$.

2026 claimed proof and new partial ranges

A February preprint by Mishra claims the full theorem for all $n\geq sk$, using sequential shifting; an overview repeats that claim, but neither supplies independent verification. Subsequent work proves substantial $4$-uniform ranges, including $n\geq 5s$ for sufficiently large $n$ and $s\geq 6961$, $n\geq 4s+4$. The August result reduces the general linear coefficient to $k+1$ and adds stability, while explicitly leaving the full conjecture unresolved outside its range.

Current status (as of August 2026): A complete proof is claimed but unverified; established results remain partial, and the general conjecture is open outside the proved ranges.