A generalized Tur\'an problem in random graphs

TL;DR

This paper investigates the generalized Turán problem in sparse random graphs, analyzing threshold phenomena for the maximum number of copies of a graph T avoiding H, revealing complex behaviors depending on graph densities.

Contribution

It extends the Erdős–Stone theorem to random graphs for certain cases and explores complex threshold behaviors when graph densities are equal or less.

Findings

Thresholds depend on densities of coverings of H with T.

Provides asymptotic behavior of the maximum copies of T avoiding H.

Identifies complex behaviors when m_2(H) ≤ m_2(T).

Abstract

We study the following generalization of the Tur\'an problem in sparse random graphs. Given graphs $T$ and $H$, let $\mathrm{ex}\big(G(n,p), T, H\big)$ be the random variable that counts the largest number of copies of $T$ in a subgraph of $G(n,p)$ that does not contain $H$. We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every $H$ and an arbitrary $2$-balanced $T$. Our results in the case when $m_2(H) > m_2(T)$ are a natural generalization of the Erd\H{o}s--Stone theorem for $G(n,p)$, which was proved several years ago by Conlon and Gowers and by Schacht; the case $T = K_m$ has been recently resolved by Alon, Kostochka, and Shikhelman. More interestingly, the case when $m_2(H) \le m_2(T)$ exhibits a more complex and subtle behavior. Namely, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of $H$ with copies of $T$ and the typical value(s) of $\mathrm{ex}\big(G(n,p), T, H\big)$ are given by solutions to deterministic hypergraph Tur\'an-type problems that we are unable to solve in full generality.