On families of faces in discrete cubes

Abstract

Let ℱ be a family ofn−k-dimensional faces of the discrete cube {0,1}ⁿ such that for allF ε ℱ, F ⊄ ∪ { F′: F ≠ F′ ∈ ℱ}. It is shown that ifn≥n ₀ (k) then |ℱ| ≤$\left( {_k^n } \right)$. This was conjectured by Aharoni and Holzman and is the casem=2 of a more general result on faces of {0,...,m−1}ⁿ.