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 (V1, V2, …, V m ) of V (H) if \(\vert S \cap V _{i}\vert \geqslant \left \lfloor \frac{\vert V _{i}\vert } {\beta (H)} \right \rfloor\) (resp. \(\vert S \cap V _{i}\vert \geqslant \left \lfloor \frac{\vert V _{i}\vert } {\beta (H)} \right \rfloor - 1\)) for all \(i\leqslant m\). 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.