Hypergraphs without exponents

Abstract

Here we give a short, concise proof for the following result. There exists a $k$-uniform hypergraph $H$ (for $k\geq 5$) without exponent, i.e., when the Tur\'an function is not polynomial in $n$. More precisely, we have $ex(n,H)=o(n^{k-1})$ but it exceeds $n^{k-1-c}$ for any positive $c$ for $n> n_0(k,c)$. This is an extension (and simplification) of a result of Frankl and the first author from 1987 where the case $k=5$ was proven. We conjecture that it is true for $k\in \{3, 4\}$ as well.