Inducibility in the hypercube

TL;DR

This paper investigates the maximum density of specific configurations within hypercubes, establishing limits and connections to graph inducibility, and determining these densities for various configurations in dimensions 3 and 4.

Contribution

It introduces a new framework for measuring configuration inducibility in hypercubes and computes these densities for several cases, extending understanding of hypercube substructure patterns.

Findings

Determined $ ext{lambda}(H,d)$ for configurations in $Q_3$ and $Q_4$.

Established connections between hypercube configurations and graph inducibility.

Provided bounds and exact values for the density of certain configurations.

Abstract

Let $Q_d$ be the hypercube of dimension $d$ and let $H$ and $K$ be subsets of the vertex set $V(Q_d)$, called configurations in $Q_d$. We say that $K$ is an \emph{exact copy} of $H$ if there is an automorphism of $Q_d$ which sends $H$ onto $K$. Let $n\geq d$ be an integer, let $H$ be a configuration in $Q_d$ and let $S$ be a configuration in $Q_n$. We let $\lambda(H,d,n)$ be the maximum, over all configurations $S$ in $Q_n$, of the fraction of sub-$d$-cubes $R$ of $Q_n$ in which $S\cap R$ is an exact copy of $H$, and we define the $d$-cube density $\lambda(H,d)$ of $H$ to be the limit as $n$ goes to infinity of $\lambda(H,d,n)$. We determine $\lambda(H,d)$ for several configurations in $Q_3$ and $Q_4$ as well as for an infinite family of configurations. There are strong connections with the inducibility of graphs.