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 nE(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.

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Correspondence to R. Ahlswede.

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Ahlswede, R., Aydinian, H. & Khachatrian, L. Maximum Number of Constant Weight Vertices of the Unit n-Cube Contained in a k-Dimensional Subspace. Combinatorica 23, 5–22 (2003). https://doi.org/10.1007/s00493-003-0011-6

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AMS Subject Classification (2000):

  • 05D05
  • 15A03