Tur\'an Densities for Small Hypercubes

TL;DR

This paper investigates the minimal size of vertex sets in high-dimensional hypercubes that intersect all smaller hypercube copies, establishing new bounds for dimensions 6 and 7, extending previous results for dimensions 3 and 4.

Contribution

It proves that the asymptotic density is strictly less than 1/(d+1) for dimensions 6 and 7, advancing understanding of Turán densities in hypercubes.

Findings

For d=7, the density is less than 1/(d+1).

For d=6, the density is less than 1/(d+1).

Extends previous results for d ≤ 4 to higher dimensions.

Abstract

How small can a set of vertices in the $n$-dimensional hypercube $Q_n$ be if it meets every copy of $Q_d$? The asymptotic density of such a set (for $d$ fixed and $n$ large) is denoted by $\gamma_d$. It is easy to see that $\gamma_d \leq 1/(d+1)$, and it is known that $\gamma_d=1/(d+1)$ for $d \leq 2$, but it was recently shown that $\gamma_d < 1/(d+1)$ for $d \geq 8$. In this paper we show that the latter phenomenon also holds for $d=7$ and $d=6$.