A short proof of a lower bound for Tur\'an numbers

TL;DR

This paper presents a concise proof establishing a lower bound for Turán numbers of strictly balanced hypergraphs, improving understanding of extremal combinatorics with elementary probabilistic methods.

Contribution

It provides a new, shorter proof for the lower bound of Turán numbers for hypergraphs, avoiding complex differential equations used in prior proofs.

Findings

Established a lower bound for Turán numbers of hypergraphs.

Introduced an elementary probabilistic proof technique.

Connected hypergraph independent sets with Turán number bounds.

Abstract

Let $F$ be a strictly balanced $r$-uniform hypergraph with $e>2$ edges and $r$-density $m$. We give a new short proof of the fact that the Tur\'an number $\ex(n, F)$ is greater than $c\, n^{r-1/m} (\log n)^{1/(e-1)}$ where $c$ depends only on $F$. The previous proof of this for $r=2$ by Bohman and Keevash and for $r \ge 3$ by Bennett and Bohman used a random greedy process and its analysis using the differential equations method. Our proof uses elementary probabilistic arguments together with a (nontrivial) classical result about independent sets in hypergraphs.