A Proof of the Erdős Matching Conjecture for Large Matchings via Sharp Tail Probability for Sums of i.i.d. Nonnegative Random Variables
Year: 2026
Type: preprint
Abstract: We resolve a conjecture of Łuczak, Mieczkowska and Šileikis on the maximal tail probability of sums of independent and identically distributed nonnegative random variables. Specifically, for every integer $k\ge1$ and every $0\le x\le1/k$, we prove \[ \sup_{\mathbb E X_1\le x}\mathbb{P}(X_1+\cdots+X_k\ge1) =\max\{1-(1-x)^k,(kx)^k\}, \] where the supremum is over nonnegative i.i.d.\ random variables $X_1,\ldots,X_k$. We also characterize all maximizing distributions. Using this result, we prove the Erdős Matching Conjecture for every fixed $k\ge3$, all sufficiently large $s$, and every $n\ge k(s+1)$. More specifically, we show that in this range, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $s+1$ pairwise disjoint edges is \[ \max\left\{ \binom{k(s+1)-1}{k}, \binom nk-\binom{n-s}{k} \right\}. \] Moreover, every extremal family consists either of all $k$-sets meeting a fixed $s$-set or of all $k$-sets contained in a fixed $(k(s+1)-1)$-set.
Source: arXiv (Cornell University)
Authors: Chi, Cheng, Sun, Shuang, Wang, Yan
Institution: Shanghai Jiao Tong University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G4716027175, G7160286548
OpenAlex ID: W7221832478