On a Class of Configuration and Coincidence Problems

We consider here a number of problems of the following general type. Let A and B be two sets in the ra-dimensional Euclidean space Em(m^ 1). B is assumed to have a center of symmetry and for any point x B(x) denotes the translate of B centered at x. An integer n(n ^ 2) is fixed and the n-fold Cartesian product A X A X · · · X A is denoted by P. If u e P then u = (xx-- , x,,) where x, e A for i = 1, · · · , n; we shall be interested in the sets B(xx), · · · , B(xn). By a configuration condition we shall understand a statement referring to the relative positions of the sets B(xJ, · · · , B(x,,) and describing their intersection properties in purely Boolean terms. Examples of admissible configuration conditions are: (i) the n sets are pairwise disjoint, (ii) their intersection is empty, (Hi) their union is connected. A configuration condition which generalizes (i) and (ii) is: an integer p is given (2 ^ p ^ n) and no p of the n sets intersect. Any admissible configuration condition C induces a partition of P into two disjoint and complementary sets Y = Y(C) and N = N(C); if u = (xx, · · · , xn) t * University of British Columbia, Vancouver. 1105