Two results about the hypercube

TL;DR

This paper advances understanding of the hypercube by precisely estimating the number of maximal VC dimension families and refining bounds on the hypercube's integrity, with implications for combinatorics and graph theory.

Contribution

It provides a tight asymptotic estimate for the number of maximal VC families and improves the upper bound on the hypercube's integrity.

Findings

Logarithm of the number of maximal VC families is asymptotically inom{n}{k} imes ext{log} n.

The upper bound on the hypercube's integrity is tightened to C rac{2^n}{ oot n} oot{ ext{log} n}.

Corollary on the asymptotics of induced matchings in the hypercube.

Abstract

First we consider families in the hypercube $Q_n$ with bounded VC dimension. Frankl raised the problem of estimating the number $m(n,k)$ of maximal families of VC dimension $k$. Alon, Moran and Yehudayoff showed that $$n^{(1+o(1))\frac{1}{k+1}\binom{n}{k}}\leq m(n,k)\leq n^{(1+o(1))\binom{n}{k}}.$$ We close the gap by showing that $\log \left(m(n,k)\right)= {(1+o(1))\binom{n}{k}}\log n$ and show how a tight asymptotic for the logarithm of the number of induced matchings between two adjacent small layers of $Q_n$ follows as a corollary. Next, we consider the integrity $I(Q_n)$ of the hypercube, defined as $$I(Q_n) = \min\{ |S| +m(Q_n \setminus S) : S \subseteq V (Q_n) \},$$ where $m(H)$ denotes the number of vertices in the largest connected component of $H$. Beineke, Goddard, Hamburger, Kleitman, Lipman and Pippert showed that $c\frac{2^n}{\sqrt{n}} \leq I(Q_n)\leq C\frac{2^n}{\sqrt{n}}\log n$ and suspected that their upper bound is the right value. We prove that the truth lies below the upper bound by showing that $I(Q_n)\leq C \frac{2^n}{\sqrt{n}}\sqrt{\log n}$.