Sidorenko Hypergraphs and Random Tur\'an Numbers

TL;DR

This paper explores the relationship between Sidorenko's conjecture and random Turán numbers in hypergraphs, establishing new bounds and characterizing specific cases like the expansion of complete hypergraphs.

Contribution

It provides new lower bounds on the maximum edges in F-free subgraphs of random hypergraphs for non-Sidorenko hypergraphs and characterizes the parameter s for certain hypergraph expansions.

Findings

Established lower bounds for non-Sidorenko hypergraphs.

Connected Sidorenko's conjecture to random Turán problems.

Determined s-value for the expansion of complete hypergraphs.

Abstract

Let $\mathrm{ex}(G_{n,p}^r,F)$ denote the maximum number of edges in an $F$-free subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$, and let $s(F):=\sup\{s: \exists H,\ t_F(H)=t_{K_r^r}(H)^{s+e(F)}>0\}$. Following recent work of Conlon, Lee, and Sidorenko, we prove non-trivial lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $s(F)>0$, i.e. $F$ is not Sidorenko. This connection between Sidorenko's conjecture and random Tur\'an problems gives new lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $s(F)>0$, and further allows us to establish upper bounds for $s(F)$ whenever upper bounds for $\mathrm{ex}(G_{n,p}^r,F)$ are known. As a consequence, we prove that $s(\mathrm{E}^r(K_{k+1}^k))=\frac{1}{r-k}$ where $\mathrm{E}^r(K_{k+1}^k)$ is the $r$-expansion of $K_{k+1}^k$.