Valuations and diameters of milnor K-rings

Given a field F and a subgroup S of F^x containing -1, we define a graph on F^x/S associated with the relative Milnor K-ring K_*^M(F)/S. We prove that if the diameter of this graph is at least 4, then there exists a valuation v on F such that S is v-open. This is done by adopting to our setting a construction in a noncommutative setting due to Rapinchuk, Segev and Seitz. We study the behavior of the diameter under important K-theoretic constructions, and relate it to the elementary type conjecture. Finally, we provide an example showing that the above bound 4 is sharp.