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Abstract: It has been conjectured that any partial triple system of order u and index λ can be embedded in a triple system of order v and index λ whenever , is even and . This conjecture is known to hold for and for all even . Here the conjecture is proven for all remaining values of λ when . [Copyright &y& Elsevier]

Abstract: A Steiner triple system of order , STS(), may be called equivalent to another STS() if one can be converted to the other by a sequence of three simple operations involving Pasch trades with a single negative block. It is conjectured that any two STS()s on the same base set are equivalent in this sense. We prove that the equivalence class containing a given system on a base set contains all the systems that can be obtained from by any sequence of well over one hundred distinct trades, and that this equivalence class contains all isomorphic copies of on . We also show that there are trades which cannot be effected by means of Pasch trades with a single negative block. [Copyright &y& Elsevier]

Abstract: This paper takes a significant step towards confirming a long-standing and far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They conjectured in 1993 that there are no non-trivial block-transitive 6-designs. We prove that the Cameron–Praeger conjecture is true for the important case of non-trivial Steiner 6-designs, i.e. for 6- designs with , except possibly when the group is with or 3, and e is an odd prime power. [Copyright &y& Elsevier]

This paper proves that a triangulated graph is an outline triple system of even index if and only if it is an amalgamated triple system of even index. 2000 Mathematical Subject Classification: primary 05B07; secondary 05C99. [ABSTRACT FROM AUTHOR]