Fair Representation by Independent Sets

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 (V 1, V 2, …, V m ) of V (H) if | S ∩ V i | ≥ | V i | β ( H ) $$\vert S \cap V _{i}\vert \geqslant \left \lfloor \frac{\vert V _{i}\vert } {\beta (H)} \right \rfloor$$ (resp. | S ∩ V i | ≥ | V i | β ( H ) − 1 $$\vert S \cap V _{i}\vert \geqslant \left \lfloor \frac{\vert V _{i}\vert } {\beta (H)} \right \rfloor - 1$$ ) for all i ≤ m $$i\leqslant m$$ . 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 K n, n . We prove that partitions of E(K n, n ) into three sets can be represented almost fairly. The methods of proofs are topological.