Erd\H{o}s Matching (Conjecture) Theorem

Let $\mathcal{F}$ be a family of $k$-sized subsets of $[n]$ that does not contain $s$ pairwise disjoint subsets. The Erd\H{o}s 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.