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

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.