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 $\mathcal{F}$ 是 $[n]$ 上所有 $k$ 元子集构成的族，且其中不含 $s$ 个两两不交的子集。极值组合学中一个著名且长期悬而未决的问题——埃尔德什匹配猜想断言，$\mathcal{F}$ 的最大基数以 $\max\left\{\binom{sk-1}{k}, \binom{n}{k}-\allowbreak \binom{n-s+1}{k}\right\}$ 为上界。这两个界限分别对应两类典型极值族的规模：一类族中所有子集均包含于某个 $sk-1$ 元基集中，另一类族中每个子集均与某个固定的 $s-1$ 元集相交。本文证明了该猜想。