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 $k$ 元子集构成的族，且不包含 $s$ 个两两不交的子集。埃尔德什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一个大小为 $sk-1$ 的基础集中，另一类是每个子集都与一个固定的 $s-1$ 元集相交。本文证明了该猜想。