The Turán Problem for Hypergraphs of Fixed Size

Abstract

We obtain a general bound on the Turán density of a hypergraph in terms of the number of edges that it contains. If ${\cal F}$ is an $r$-uniform hypergraph with $f$ edges we show that $$\pi({\cal F}) < {f-2\over f-1} - \big(1+o(1)\big)(2r!^{2/r}f^{3-2/r})^{-1},$$ for fixed $r \geq 3$ and $f \rightarrow \infty$.