Extremal numbers and Sidorenko's conjecture

TL;DR

This paper explores the limitations of Sidorenko's conjecture in hypergraphs, showing that its failure can lead to improved bounds on extremal numbers and providing new counterexamples for certain hypergraph classes.

Contribution

It demonstrates that failure of Sidorenko's conjecture in hypergraphs can be used to improve extremal number bounds and identifies new counterexamples including linear hypergraphs with loose triangles and 3-uniform tight cycles.

Findings

Sidorenko's conjecture does not hold for certain hypergraphs.

Failure of the conjecture allows improved bounds on extremal numbers.

New counterexamples include hypergraphs with loose triangles and tight cycles.

Abstract

Sidorenko's conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko's conjecture does not hold for a particular $r$-partite $r$-uniform hypergraph $H$, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number $\mathrm{ex}(n,H)$, the maximum number of edges in an $n$-vertex $H$-free $r$-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all $3$-partite $3$-uniform tight cycles.