On a covering problem in the hypercube

TL;DR

This paper investigates a specific covering problem in the hypercube related to Turán and Ramsey questions, providing bounds and constructions for the minimal set size and coloring constraints in high-dimensional hypercubes.

Contribution

It offers new bounds and explicit constructions for covering and coloring problems involving subcubes within hypercubes, advancing understanding of combinatorial structures.

Findings

Established upper and lower bounds for the covering set size

Provided constructions for specific cases of the covering problem

Analyzed coloring constraints in hypercube substructures

Abstract

In this paper, we address a particular variation of the Tur\'an problem for the hypercube. Alon, Krech and Szab\'o (2007) asked "In an n-dimensional hypercube, Qn, and for l < d < n, what is the size of a smallest set, S, of Q_l's so that every Q_d contains at least one member of S?" Likewise, they asked a similar Ramsey type question: "What is the largest number of colors that we can use to color the copies of Q_l in Q_n such that each Q_d contains a Q_l of each color?" We give upper and lower bounds for each of these questions and provide constructions of the set S above for some specific cases.