A short proof of Erdős’ conjecture for triple systems

Abstract

Erdős [1] conjectured that for all ${k \geq 2}$, ${s \geq 1}$ and ${n \geq {k(s+1)}}$, an n-vertex k-uniform hypergraph ${\mathcal{F}}$ with ${\nu(\mathcal{F})=s}$ cannot have more than ${\max\{\binom{sk+k-1}k,\binom nk-\binom{n-s}k\}}$ edges. It took almost fifty years to prove it for triple systems. In [5] we proved the conjecture for all s and all ${n \geq 4(s+1)}$. Then Łuczak and Mieczkowska [6] 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 Erdős’ conjecture for ${s \geq 33}$. Our motivation is to lay down foundations for a possible proof in the much harder case k = 4, at least for large s.