Geometric systems of unbiased representatives

Geometric Systems of Disjoint Representatives

Consider a finite collection of subsets of a metric space and ask for a system of representatives which are pairwise at a distance at least q, where q is a parameter of the problem. In discrete spaces this generalizes the well known problem of distinct representatives, while in Euclidean metrics the problem reduces to finding a system of disjoint balls. This problem is closely related to practi...

System of unbiased representatives for a collection of bicolorings

Let B denote a set of bicolorings of [n], where each bicoloring is a mapping of the points in [n] to {−1,+1}. For each B ∈ B, let YB = (B(1), . . . , B(n)). For each A ⊆ [n], let XA ∈ {0, 1} denote the incidence vector of A. A non-empty set A is said to be an ‘unbiased representative’ for a bicoloring B ∈ B if 〈XA, YB〉 = 0. Given a set B of bicolorings, we study the minimum cardinality of a fam...

Systems of sets and their representatives

Systems of Distinct Representatives and Linear Algebra *

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Cooperative Colorings and Independent Systems of Representatives

Two closely related notions (in fact, translatable into one another), are those of cooperative colorings and of independent systems of representatives (ISRs). A cooperative coloring of a family of (not necessarily distinct) graphs G1, G2, . . . , Gk on the same vertex set V is a choice of independent sets Ai in Gi (1 ≤ i ≤ k) such that ⋃k i=1 Ai = V . An ISR for a partition of the vertex set of...