Generalized intersection patterns and two-symbol balanced arrays

Abstract Let m be a fixed integer, m = {0,1,⋯, m − 1}; let C be a family of nonvoid subsets of m, and let R be a hereditary subfamily of C . Given finite sets A m ,…, A m −1 such that ∩ iϵB A i = O for all B ⊆ m , B ∉ C , the vector of |∩ iϵR A i | ( R j ϵ R ) is called a C -supported R -intersection pattern. The characterization of the Y RC of such patterns is a difficult combinatorial problem even for m =5 and simple families R and C . We study the algebraic structure of the convex cone Y RC and its dual, and an integer linear-programming aspect of the problem; in particular we introduce the notion of content and pseudocontent. A relaxation leads to quadratic and higher forms over certain subsets of reals. As an application we study the natural link between highly symmetric patterns and two-symbol balanced arrays.