Generalized Tur\'an densities in the hypercube

TL;DR

This paper explores generalized Turán densities in hypercubes, focusing on the maximum number of smaller hypercubes or cycles within subgraphs that avoid certain substructures.

Contribution

It extends Turán-type extremal problems to hypercubes, analyzing the maximum counts of specific subgraphs without containing others.

Findings

Determined bounds for generalized Turán densities in hypercubes.

Characterized extremal configurations for avoiding certain subgraphs.

Provided asymptotic estimates for subgraph counts in hypercube settings.

Abstract

A classical extremal, or Tur\'an-type problem asks to determine ${\rm ex}(G, H)$, the largest number of edges in a subgraph of a graph $G$ which does not contain a subgraph isomorphic to $H$. Alon and Shikhelman introduced the so-called generalized extremal number ${\rm ex}(G,T,H)$, defined to be the maximum number of subgraphs isomorphic to $T$ in a subgraph of $G$ that contains no subgraphs isomorphic to $H$. In this paper we investigate the case when $G = Q_n$, the hypercube of dimension $n$, and $T$ and $H$ are smaller hypercubes or cycles.