TOPOLOGICAL BOUNDS FOR GRAPH REPRESENTATIONS OVER ANY FIELD

Haviv [European J. Combin., 81 (2019), pp. 84-97] has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over R. We show that this actually holds for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over R --an important graph invariant from coding theory-and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed.