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Abstract: Sylvester’s problem, i.e., the problem of the existence of large sets of Kirkman triple systems (LKTSs), is one of the most challenging problems in combinatorial design theory. Most recursive constructions for LKTSs impose resolvable restrictions on the ingredient designs, making the recursions difficult to apply. In this note we provide necessary small designs by using spherical geometries and a direct construction. Finally, we prove the existence of LKTS for all positive integers . A new result on overlarge sets of Kirkman triple systems is also produced. [Copyright &y& Elsevier]

Abstract: The problem of the existence of large sets of Kirkman triple systems (LKTS) is one of the most celebrated open problems in design theory. Only a few sparsely distributed infinite classes have been determined, although LKTS have been investigated by many authors. The purpose of this paper is to survey constructions and results on LKTS and related designs. A systematic account of this work is provided. Most of the known constructions are unified and generalized; the approaches to LKTS are enriched; a couple of new constructions for related designs are also displayed. In particular, a new existence class of LKTS is demonstrated and some new results of overlarge sets of Kirkman triple systems are also produced. [Copyright &y& Elsevier]

Abstract: An () over a -set is a collection of pairwise disjoint resolvable (almost resolvable) Mendelsohn triple systems of order . In this paper several direct construction methods for OLRMTSs and OLARMTSs are presented and then applied to produce some new orders; the smallest unknown for , the smallest unknown for , and some other small designs are displayed; a few new existence families are also obtained by known recursive constructions. [Copyright &y& Elsevier]

Abstract: The concept of good large set of Steiner triple systems (or GLS in short) was introduced by Lu in his paper “on large sets of disjoint Steiner triple systems”, [J. Lu, On large sets of disjoint Steiner triple systems, I–III, J. Combin. Theory (A) 34 (1983) 140-182]. In this paper a doubling construction for GLSs is displayed and some existence results are obtained. [Copyright &y& Elsevier]

Abstract: An idempotent quasigroup of order is called resolvable (denoted by RIQ) if the set of non-idempotent 3-vectors can be partitioned into disjoint transversals. An overlarge set of idempotent quasigroups of order , briefly by OLIQ, is a collection of IQs, with all the non-idempotent 3-vectors partitioning all those on a -set. An OLRIQ is an OLIQ with each member IQ being resolvable. In this paper, it is established that there exists an OLRIQ for any positive integer , except for , and except possibly for . An OLIQ is another type of restricted OLIQ in which each member IQ has an idempotent orthogonal mate. It is shown that an OLIQ exists for any positive integer , except for , and except possibly for . [Copyright &y& Elsevier]