  In this paper we study the spaces of $q$-tuples of points in a Euclidean
space, without $k$-wise coincidences (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.
