Extremal Numbers and Sidorenko’s Conjecture

Author:

Conlon David1,Lee Joonkyung2,Sidorenko Alexander3

Affiliation:

1. Department of Mathematics, California Institute of Technology , Pasadena, CA 91125, USA

2. Department of Mathematics, Yonsei University , 03722 Seoul, South Korea

3. Department of Extremal Combinatorics, Alfréd Rényi Institute of Mathematics , H-1364 Budapest, Hungary

Abstract

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 $\textrm{ex}(n,H)$, 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.

Publisher

Oxford University Press (OUP)

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Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Correction to: Extremal Numbers and Sidorenko’s Conjecture;International Mathematics Research Notices;2024-06-11

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