Short Proof of Erd\H os Conjecture for Triple Systems

In 1965 Erd\H os conjectured that for all $k\ge 2$, $s\ge 1$ and $n\ge k(s+1)$, an $n$-vertex $k$-uniform hypergraph $\text{F}$ with $\nu (\text{F})=s$ cannot have more than \newline $\max \{\left(\genfrac{}{}{0px}{}{sk+k-1}{k}\right),\left(\genfrac{}{}{0px}{}{n}{k}\right)-\left(\genfrac{}{}{0px}{}{n-s}{k}\right)\}$ edges. It took almost fifty years to prove it for triple systems. In 2012 we proved the conjecture for all $s$ and all $n\ge 4(s+1)$. 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 $s\ge 33$. Our motivation is to lay down foundations for a possible proof in the much harder case $k=4$, at least for large $s$.