A Note on the Subcubes of the $n$-Cube

TL;DR

This paper simplifies existing proofs regarding the optimal selection of vertices in the n-cube to maximize q-dimensional subcubes, and applies Graham's lemma to solve a related recursive equation, enhancing understanding of combinatorial structures.

Contribution

It provides simplified proofs for a known combinatorial problem and introduces a new application of Graham's lemma to solve a related recursive equation.

Findings

Simplified proof of the optimal vertex set in the n-cube for maximizing q-subcubes.

Application of Graham's lemma to streamline proof techniques.

Solution to a recursive equation related to the optimization problem.

Abstract

In the year 1990, B\'ela Bollob\'as, Imre Leader and Andrew Radcliffe considered the following combinatorial problem: given three parameters k, n and q, find a set of k vertices in the binary n-cube which contains a maximal number of q-dimensional subcubes. It was shown that an optimal solution is given by the k vertices which coincide with the binary representations of the number 0 , 1 , ... , k-1. Two proofs were presented. The proof given by Bollobas and Leader is particularly elegant and short. Here we show that also the other proof, the one given by Bollobas and Radcliffe, becomes quite simple and short when it is combined with a lemma from Graham whose publication dates back to 1970. As a second application of Graham's lemma, we solve a recursive equation (related to the optimization problem that we discussed before) that might be considered interesting in its own right.