Maximal number of constant weight vertices of the unit n-cube contained in a k-dimensional subspace

Summary form only given. We introduce and solve a seemingly basic geometrical extremal problem. The set of vertices of weight w in the unit cube of Rⁿ, E(n,w)={xⁿ∈{0,1} ⁿ:xⁿ has w ones} can also be viewed as the set in which constant weight codes are studied in information theory. Another interest there is in linear codes. This was a motivation for studying the interplay between two properties: constant weight and linearity. In particular we wanted to know M(n,k,w)^Δ₌max{|U_kⁿ ∩E(n,w)|: U_kⁿ is a k-dimensional linear subspace of Rⁿ}, that is, the maximal cardinality of a set of vectors in E(n,w), whose linear span has a dimension not exceeding k. Our complete solution is given. We also present an extension to multi-sets and explain a connection to the (simpler) Erdos-Moser problem