Short Proof of Erd\H os Conjecture for Triple Systems

TL;DR

This paper presents a simplified proof of Erdős's conjecture for triple systems, establishing the maximum number of edges in hypergraphs with a given matching number for certain parameters, and aims to inform future work on higher uniformities.

Contribution

The authors provide a simpler proof of Erdős's conjecture for triple systems, covering cases where s ≥ 33, and set groundwork for tackling the conjecture for 4-uniform hypergraphs.

Findings

Proof of Erdős's conjecture for triple systems with s ≥ 33.

Established maximum edge bounds for hypergraphs with given matching number.

Simplified proof approach to facilitate future research on higher uniformities.

Abstract

In 1965 Erd\H os conjectured that for all $k\ge2$, $s\ge1$ and $n\ge k(s+1)$, an $n$-vertex $k$-uniform hypergraph $\F$ with $\nu(\F)=s$ cannot have more than \newline $\max\{\binom{sk+k-1}k,\;\binom nk-\binom{n-s}k\}$ edges. It took almost fifty years to prove it for triple systems. In 2012 we proved the conjecture for all $s$ and all $n\ge4(s+1)$. Then {\L}uczak and Mieczkowska (2013) proved the conjecture for sufficiently large $s$ and all $n$. Soon after, Frankl proved it for all $s$. Here we present a simpler version of that proof which yields Erd\H os's conjecture for $s\ge33$. Our motivation is to lay down foundations for a possible proof in the much harder case $k=4$, at least for large $s$.