The spectrum of \(\alpha\)-resolvable block designs with block size 3 (Q1183993)

A balanced incomplete block design \(D(v,k,\lambda)\) is called \(\alpha\)- resolvable if its blocks can be partitioned into classes such that each point of the design lies in exactly \(\alpha\) blocks of each class. Necessary conditions for such a partitioning to exist include (i) \(bk=vr\); (ii) \(\lambda (v-1)=r(k-1)\); (iii) \(k\mid \alpha v\); (iv) \(\alpha\mid r\), where (i) and (ii) are the usual necessary conditions for the existence of any BIBD. The present paper is concerned with the sufficiency of the above conditions in the case when \(k=3\) (i.e., triple systems). When \((v,\alpha)=(6,1)\), \textit{A. Hartman} [A survey of the existence problem for resolvable designs, Proc. 1st Franco-southeast Asian math. Conf., Singapore 1979, Vol. 2, Southeast Asian Bull. Math., Special Issue, 268- 277 (1979; Zbl 0444.05019)] has shown that a 1-resolvable \((6,3,\lambda)\)-design exists if and only if \(\lambda\equiv 0\pmod 4\). If \((v,\alpha)\neq (6,1)\), the authors of this article show that conditions (i)--(iv) above are sufficient for the existence of an \(\alpha\)- resolvable \((v,3,\lambda)\)-design. One of the main techniques used is the notion of a resolvable frame as introduced by \textit{H. Hanani} in [On resolvable balanced incomplete block designs, J. Comb. Theory, Ser. A. 17, 275-289 (1974; Zbl 0305.05010)].