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

We introduce and solve a natural geometrical extremal problem. For the set E (n,w) = {x ⁿ ∈ {0,1}ⁿ : x ⁿ has w ones } of vertices of weight w in the unit cube of ℝⁿ we determine M (n,k,w) ≜ max{|U _k ⁿ ∩ E(n,w)|:U _k ⁿ 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.