Coloring by two-way independent sets

The paper stems from an attempt to investigate a somewhat mysterious phenomenon: conditions which suffice for the existence of a “large” set satisfying certain conditions (e.g., a large independent set in a graph) often suffice (or at least are conjectured to suffice) for the existence of a covering of the ground set by few sets satisfying these conditions (in the example of independent sets in a graph this means that the graph has small chromatic number). We consider two conjectures of this type, on coloring by sets which are “two-way independent”, in the sense of belonging to a matroid and at the same time being independent in a graph sharing its ground set with the matroid. We prove these conjectures for matroids of rank 2. We also consider dual conjectures, on packing bases of a matroid, which are independent in a given graph.