Rational exponents for hypergraph Turan problems

Given a family of $k$-hypergraphs $\mathcal{F}, ex(n,\mathcal{F})$ is the maximum number of edges a $k$-hypergraph can have, knowing that said hypergraph has $n$ vertices but contains no copy of any hypergraph from $\mathcal{F}$ as a subgraph. We prove that for a rational $r$, there exists some finite family $\mathcal{F}$ of $k$-hypergraphs for which $ex(n,\mathcal{F}) = \Theta (n^{k-r})$ if and only if $0 \leq r \leq k-1$ or $r=k$.

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Received 15 June 2017

Published 7 December 2018