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Abstract: We give the first known examples of 6-sparse Steiner triple systems by constructing 29 such systems in the residue class 7 modulo 12, with orders ranging from 139 to 4447. We then present a recursive construction which establishes the existence of 6-sparse systems for an infinite set of orders. Observations are also made concerning existing construction methods for perfect Steiner triple systems, and we give a further example of such a system. This has order 135,859 and is only the fourteenth known. Finally, we present a uniform Steiner triple system of order 180,907. [Copyright &y& Elsevier]

Abstract: In [Z. Füredi, Turán type problems, in: Surveys in Combinatorics, Guildford, 1991, in: London Math. Soc. Lecture Note Ser., vol. 166, Cambridge Univ. Press, Cambridge, 1991, pp. 253–300, MR1161467 (93d:05072)], Füredi raised a conjecture about the maximum size of L-intersecting families. In this note, we address a variant of this conjecture. In particular, we show that for any Steiner triple system on , there exist a family of k-sets on with and such that for every the family is isomorphic to . [Copyright &y& Elsevier]

Abstract: It is known that the Horton starters can be used to construct uniform one-factorizations of the complete graph. Of primary interest is the cycle structure of such one-factorizations. In this paper we give some general conditions for the existence of k-cycles, then specialize this to the cases , completely characterizing the four-cycle case. We also show that for each even and any positive integer N there exists a uniform one-factorization in some large enough complete graph containing at least N k-cycles. [Copyright &y& Elsevier]

Abstract: A Steiner triple system of order (briefly STS) consists of a -element set X and a collection of 3-element subsets of X, called blocks, such that every pair of distinct points in X is contained in a unique block. A large set of disjoint STS (briefly LSTS) is a partition of all 3-subsets (triples) of X into STS. In 1983–1984, Lu Jiaxi first proved that there exists an LSTS for any or with six possible exceptions and a definite exception . In 1989, Teirlinck solved the existence of LSTS for the remaining six orders. Since their proof is very complicated, it is much desired to find a simple proof. For this purpose, we give a new proof which is mainly based on the 3-wise balanced designs and partitionable candelabra systems. [Copyright &y& Elsevier]

We prove that for n>3 every STS(n) has both an orientable and a nonorientable embedding in which the triples of the STS(n) appear as triangular faces and there is just one additional large face. We also obtain detailed results about the possible automorphisms of such embeddings. [Copyright &y& Elsevier]

Let q be 2m with m≥1. We construct a family of dual hyperovals over GF(q) inside the 2n-1-dimensional projective space PG(GF(qn)×GF(qn)). [Copyright &y& Elsevier]

In Adv. Geom. 3 (2003) 245, a class of d-dimensional dual hyperovals is constructed starting from a subset X of PG(d,2) with certain properties. In this paper, a criterion for X to provide a d-dimensional dual hyperoval is given in terms of some functions. Based on this, we describe such subsets, and show that there are exactly two isomorphism classes of d-dimensional dual hyperovals arising from those subsets and that a similar statement holds for those of the associated Steiner systems S(3,4,2d+1). [Copyright &y& Elsevier]

Let S be a Steiner triple system and G a cubic graph. We say that G is S-colourable if its edges can be coloured so that at each vertex the incident colours form a triple of S. We show that if S is a projective system PG(n,2), n⩾2, then G is S-colourable if and only if it is bridgeless, and that every bridgeless cubic graph has an S-colouring for every Steiner triple system of order greater than 3. We establish a condition on a cubic graph with a bridge which ensures that it fails to have an S-colouring if S is an affine system, and we conjecture that this is the only obstruction to colouring any cubic graph with any non-projective system of order greater than 3. [Copyright &y& Elsevier]