Systems of distant 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...

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...

Transversals, systems of distinct representatives, mechanism design, and matching

A transversal generated by a system of distinct representatives (SDR) for a collection of sets consists of an element from each set (its representative) such that the representative uniquely identifies the set it belongs to. Theorem 1 gives a necessary and sufficient condition that an arbitrary collection, finite or infinite, of sets, finite or infinite, have an SDR. The proof is direct, short....