Pairwise balanced designs PBD\((v,K)\) with \(3\in K\) (Q2716622)

A pairwise balanced design (PBD) is a finite incidence structure of points and blocks such that any two points are incident with exactly \(\lambda=1\) blocks. Only the case \(\lambda=1\) is considered. Another name for such a PBD is linear space. A \(\text{PBD}(v, K)\) is a PBD with all block sizes in the subset \(K\) of the set \(\mathbb{N}\) of natural numbers (here with 3 in \(K\) and \(1\), \(2\) not in \(K\)).NEWLINENEWLINENEWLINEThe following necessary existence conditions for a \(\text{PBD}(v, K)\) are well known. Let \(\alpha\) be the greatest common divisor (GCD) of all \(k-1\) and \(\beta\) the GCD of all \(k(k-1)\) with \(k\) in \(K\). Then, for any \(\text{PBD}(v, K)\), \(k-1\) must divide \(v-1\) and \(k(k-1)\) must divide \(v(v-1)\).NEWLINENEWLINENEWLINEIn one of his famous papers R. W. Wilson proved that these necessary conditions are asymptotically sufficient. That is: For every \(K\) there is a number \(v_0(K)\) such that the necessary conditions together with \(v>v_0(K)\) are sufficient for the existence of a \(\text{PBD}(v, K)\). In general these numbers \(v_0(K)\) are frightfully large. In the present paper the authors succeed in finding reasonably low values of them if \(3\) is in \(K\).NEWLINENEWLINENEWLINEIn Section 2 they restate former results of several authors. In particular they use the concept of MRD (mandatory representation design or Mendelsohn-Rees design). In Section 3 they give and prove their Theorem 3.1, and in Section 4 they develop the relevant constructions. In two tables they collect their results, with surprisingly low values of the numbers \(v_0(K)\).NEWLINENEWLINENEWLINEIn the references the abbreviation [G1], first time, should be replaced by [GMP].