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

TL;DR

This paper proves the Erd ext{"o}s Matching Conjecture, a fundamental open problem in extremal combinatorics, establishing the maximum size of certain families of sets without s disjoint subsets.

Contribution

The paper provides a proof of the Erd ext{"o}s Matching Conjecture, confirming the conjectured upper bounds for the size of families of sets avoiding s disjoint members.

Findings

Confirmed the conjectured upper bounds for the maximum size of set families.

Established the extremal structures achieving these bounds.

Resolved a long-standing open problem in extremal combinatorics.

Abstract

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.