Fair representation by independent sets

Abstract

For a hypergraph H let β(H) denote the minimal number of edges from H covering V(H). An edge S of H is said to represent fairly (resp. almost fairly) a partition (V1, V2, …, Vm) of for all i ≤ m. In matroids any partition of V(H) can be represented fairly by some independent set. We look for classes of hypergraphs H in which any partition of V(H) can be represented almost fairly by some edge.We show that this is true when H is the set of independent sets in a path, and conjecture that it is true when H is the set of matchings in Kn;n. We prove that partitions of E(Kn;n) into three sets can be represented almost fairly. The methods of proofs are topological.