The motif problem: Geometric representations of sets of equivalence relations

∗ (H) . We obtain more precise optimization results for two special types of motifs (uniform motifs and single-starred motifs). For uniform motifs, we show that our upper bound can be attained and characterize the subsets (called grids) which achieve the upper bound. We prove similar results for single-starred motifs, in which case sufficiently large subsets containing the maximum number of motifs must be contained in a union of a predetermined number of lines, the number depending only on the motif specification.