Let $\mathcal{F}$ be a family of $k$-sized subsets of $[n]$ that does not contain $s$ pairwise disjoint subsets. The Erdős Matching Conjecture, a celebrated and long-standing open problem in extremal combinatorics, asserts the maximum cardinality of $\mathcal{F}$ is upper bounded by $\max\left\{\binom{sk-1}{k}, \binom{n}{k}-\allowbreak \binom{n-s+1}{k}\right\}$. These two bounds correspond to the sizes of two canonical extremal families: one in which all subsets are contained within a ground set of $sk-1$ elements, and one in which every subset intersects a fixed set of $s-1$ elements. In this paper, we prove the conjecture.

翻译：设 $\mathcal{F}$ 为 $[n]$ 上 $k$ 元子集族，且不包含 $s$ 个两两不交的子集。Erdős匹配猜想作为极值组合学中一个著名且长期悬而未决的开放问题，断言 $\mathcal{F}$ 的最大基数上界为 $\max\left\{\binom{sk-1}{k}, \binom{n}{k}-\allowbreak \binom{n-s+1}{k}\right\}$。这两个上界分别对应两类典型极值族的规模：一类是所有子集均包含于大小为 $sk-1$ 的基础集中，另一类是每个子集都与某个固定的 $s-1$ 元集相交。本文证明了该猜想。