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Source. Z. Füredi and D. Gerbner, Hypergraphs without exponents, J. Combin. Theory Ser. A 184 (2021), Paper No. 105517, doi:10.1016/j.jcta.2021.105517. Labels and pages are those of the arXiv preprint arXiv:1906.06657v1 named on the source card.

Statement

Conjecture (p. 2, quoted). "We conjecture that examples with no exponents should exist for $k=3$ and 4, too."

An example here is a single $k$-uniform hypergraph $H$ for which there is no real $\alpha$ with $\mathrm{ex}_k(n,H)=\Theta(n^\alpha)$ (p. 2). The abstract (p. 1) states the same conjecture for $k\in\{3,4\}$. For $k\ge5$ such hypergraphs are given by Theorem 3.3, for instance $Q_k(3)$. The paper recalls (p. 2) that the Ruzsa--Szemerédi $(6,3)$-theorem gives a family of two 3-uniform hypergraphs with no exponent, so the open case is that of a single forbidden hypergraph.

Proof pointer

Posed without proof.

Read depth

Claims checked: read on the page images of pp. 1--2 of the preprint.

Bears on

• Problem 713: the conjecture concerns 3- and 4-uniform hypergraphs, not graphs, and the paper relates it to the problem in no way beyond recalling the Erdős--Simonovits conjectures on graph exponents (pp. 1--2).