Irredundant Families of Subcubes

TL;DR

This paper investigates the maximum size of irredundant families of k-dimensional subcubes within the n-cube, providing new bounds and proofs that advance understanding of their combinatorial structure.

Contribution

The paper offers a new proof of Meshulam's upper bound and establishes tight bounds for specific irredundant families, advancing the theoretical understanding of subcube arrangements.

Findings

Meshulam's upper bound is tight up to a factor of e.

For k >= n/2, irredundant families passing through fixed points have size at most {n choose k}.

The lower bound matches the upper bound up to a constant factor.

Abstract

We consider the problem of finding the maximum possible size of a family of k-dimensional subcubes of the n-cube {0,1}^{n}, none of which is contained in the union of the others. (We call such a family `irredundant'). Aharoni and Holzman conjectured that for k > n/2, the answer is {n choose k} (which is attained by the family of all k-subcubes containing a fixed point). We give a new proof of a general upper bound of Meshulam, and we prove that for k >= n/2, any irredundant family in which all the subcubes go through either (0,0,...,0) or (1,1,...,1) has size at most {n choose k}. We then give a general lower bound, showing that Meshulam's upper bound is always tight up to a factor of at most e.