In this paper the spaces of $q$-tuples of points in a Euclidean space,
without $k$-wise coincidences are studied (configuration-like spaces). A
transitive group action by permuting these points is considered, and some new
upper bounds on the genus (in the sense of Krasnosel'skii-Schwarz and
Clapp-Puppe) for this action are given. Some theorems of Cohen-Lusk type for
coincidence points of continuous maps to Euclidean spaces are deduced.