Maximum Number of Constant Weight Vertices of the Unit n -Cube Contained in a k -Dimensional Subspace

Abstract

We introduce and solve a natural geometrical extremal problem. For the set E(n,w) = {xn ∈ {0,1}n : xn has w ones } of vertices of weight w in the unit cube of ℝn we determine M (n,k,w) ≜ max{|Ukn ∩ E(n,w)|:Ukn is a k-dimensional subspace of ℝn. We also present an extension to multi-sets and explain a connection to a higher dimensional Erdős–Moser type problem.