The motif problem

Fix a choice and ordering of four pairwise non-adjacent vertices of a parallelepiped, and call a motif a sequence of four points in R 3 that coincide with these vertices for some, possibly degenerate, parallelepiped whose edges are parallel to the axes. We show that a set of r points can contain at most r 2 motifs. Generalizing the notion of motif to a sequence of L points in R p , we show that the maximum number of motifs that can occur in a point set of a given size is related to a linear programming problem arising from hypergraph theory, and discuss some related questions.