The exact block-type spectrum in the Colbourn–Dinitz–Rosa problem

In 1999 Colbourn, Dinitz and Rosa asked which three class sizes can occur. We ask a further question about the blocks of each such system. Fix three color classes in a Steiner triple system and require each block to meet exactly two of them. Of the six block-type counts, any one determines the other five. For sufficiently large admissible profiles, we characterize the possible values of the number of blocks with two points in the first class and one in the third. An integer occurs exactly when it passes two arithmetic tests obtained by counting pairs through a point. The proof uses the typed absorption theorems T and T+ of Triangle decompositions III [1]. In the interior T+ gives an approximate count and we select a set of reserved nine-point exchanges, each worth one unit. At an endpoint we place the scarce types first and apply T to the remaining pairs. Existence for every admissible order is proved separately in [2]. English manuscript, Spanish translation, shared Lean companion, frozen sources and publication evidence. Publication repository. Manuscripts, companion and original figures: CC BY 4.0. Bundled Lean code retains its upstream licenses.