Random Tur\'an Problems for Hypergraph Expansions

TL;DR

This paper studies the maximum size of hypergraphs avoiding certain expansions within random hypergraphs, revealing phase transitions for large uniformities and unifying previous results in the field.

Contribution

It provides the first systematic analysis of the Turán problem for hypergraph expansions in random hypergraphs, especially for large uniformities and Sidorenko hypergraphs.

Findings

Identifies three phase transitions in the Turán number for large uniformities.

Shows more complex behavior for non-Sidorenko hypergraph expansions.

Unifies and extends previous results on random Turán problems for hypergraphs.

Abstract

Given an $r_0$-uniform hypergraph $F$, we define its $r$-uniform expansion $F^{(r)}$ to be the hypergraph obtained from $F$ by inserting $r-r_0$ distinct vertices into each edge of $F$, and we define $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ to be the largest $F^{(r)}$-free subgraph of the random hypergraph $G_{n,p}^r$. We initiate the first systematic study of $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ for general hypergraphs $F$. Our main result essentially resolves this problem for large $r$ by showing that $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ goes through three predictable phases whenever $F$ is Sidorenko and $r$ is sufficiently large, with the behavior of $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ being provably more complex whenever $F$ has no Sidorenko expansion. Moreover, our methods unify and generalize almost all previously known results for the random Tur\'an problem for degenerate hypergraphs of uniformity at least 3.