  Combining Ky Fan's theorem with ideas of Greene and Matousek we prove a
generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam
theorem due to Bacon and Tucker, we also prove the presence of all possible
completely multicolored t-vertex complete bipartite graphs in t-colored
t-chromatic Kneser graphs and in several of their relatives. In particular,
this implies a generalization of a recent result of G. Spencer and F. E. Su.
