Erdős Matching (Conjecture) Theorem
Tapas Kumar Mishra
TL;DR
The paper resolves the Erdős Matching Conjecture by proving that for all $n \ge sk$ and $k, s \ge 1$, any $k$-uniform family $\mathcal{F} \subseteq \binom{[n]}{k}$ with no $s$ pairwise disjoint members satisfies $|\mathcal{F}| \le \max\left\{\binom{n}{k}-\binom{n-s+1}{k},\binom{sk-1}{k}\right\}$. The authors develop an algorithmic shifting framework that uses Fránkl's $(i,j)$ shift and a novel Multiple Sequential Shift operator $\mathcal{S}_{i_1j_1,\ldots,i_rj_r}$, coupled with a potential function $\Phi(\mathcal{G})$ counting sets intersecting a fixed $S$ of size $s-1$, to steer any valid family toward a canonical extremal structure. The proof identifies a dichotomy between trivial and non-trivial families and proves that shift-based operations preserve total size while not increasing the matching number, leading to a stable configuration and the extremal bounds. This completes a central open problem in extremal set theory and opens avenues for stability results, rainbow analogues, and algorithmic insights into hypergraph matching.
Abstract
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.