The Erdős Matching Conjecture for 4-uniform hypergraphs

Abstract

We prove the Erdős Matching Conjecture for $4$-uniform hypergraphs for every matching number $s\ge 6004$. Specifically, for integers $s\ge 6004$ and $n\ge 4s+4$, if a $n$-vertex 4-uniform hypergraph $\mathcal{F}$ does not contain a matching of size $s+1$, then \[ |\mathcal F|\le \max\left\{\binom{4s+3}{4},\binom n4-\binom{n-s}{4}\right\}. \] Our proof introduces a finite-board reduction for general uniformity. It reduces the global conjecture to a lower-uniformity bound and a fixed finite optimization at the two adjacent vertex numbers where the candidate constructions exchange dominance. In the $4$-uniform case the board has $19$ vertices, and its weighted inequality splits into $19$-, $15$-, and $11$-vertex layers. The hardest layer is resolved by exact rational dual certificates and deterministic integer searches. All computer-assisted steps are checked in exact arithmetic by verifiers that reconstruct the finite systems directly from their mathematical definitions.