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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: An LRMTS is a large set consisting of pairwise disjoint resolvable Mendelsohn triple systems defined over the same -element set. In this paper a new product construction for LRMTS is displayed by using generalized LR-designs. As well, several new existence families of LRMTS are constructed via 3-wise balanced designs. Finally the existence result of LRMTS is expanded by combining all the known constructions. [Copyright &y& Elsevier]

Abstract: An idempotent Latin square of order is called resolvable and denoted by RILS if the off-diagonal cells can be resolved into disjoint transversals. A large set of resolvable idempotent Latin squares of order , briefly LRILS, is a collection of RILSs pairwise agreeing on only the main diagonal. In this paper we display some recursive and direct constructions for LRILSs. [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 is a collection of disjoint pure Mendelsohn triple systems on the same set of elements. In this paper, the concept of t-purely partitionable Mendelsohn candelabra system (or t-PPMCS in short) is introduced for constructing . A powerful recursive construction for t-PPMCSs is also displayed by utilizing s-fan designs. Together with direct constructions, the existence of an for and is established. For odd integer , a special construction from both and to is set up. Finally, the existence of an is completely determined to be the set . [Copyright &y& Elsevier]

A Mendelsohn triple system MTS (v , b) is a collection of b cyclic triples (blocks) on a set of v points. It is j -balanced for j = 1 , 2 , 3 when any two points, ordered pairs, or cyclic triples (resp.) are contained in the same or almost the same number of blocks (difference at most one). A (2 , 3) -balanced Mendelsohn triple system is an MTS (v , b) that is both 2-balanced and 3-balanced. Employing large sets of Mendelsohn triple systems and partitionable Mendelsohn candelabra systems, we completely determine the spectrum for which a 2-balanced Mendelsohn triple system exists. Meanwhile, we determine the existence spectrum for a (2 , 3) -balanced Mendelsohn triple system. [ABSTRACT FROM AUTHOR]