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A triple system is cancellative if it does not contain three distinct sets A , B , C such that the symmetric difference of A and B is contained in C. We show that every cancellative triple system H that satisfies a particular inequality between the sizes of H and its shadow must be structurally close to the balanced blowup of some Steiner triple system. Our result contains a stability theorem for cancellative triple systems due to Keevash and Mubayi as a special case. It also implies that the boundary of the feasible region of cancellative triple systems has infinitely many local maxima, thus giving the first example showing this phenomenon. [ABSTRACT FROM AUTHOR]

With each Steiner triple system there is associated a one-parameter family of commutative, nonassociative, nonunital algebras that are by construction exact, meaning that the trace of every multiplication operator vanishes, and these algebras are shown to be Killing metrized, meaning the Killing type trace-form is nondegenerate and invariant (Frobenius), and simple, except for certain parameter values. The definition of these algebras resembles that of the Matsuo algebra of the Steiner triple system, but they are different. For a Hall triple system, the associated algebra is a primitive axial algebra for a Z / 2 Z -graded fusion law. [ABSTRACT FROM AUTHOR]

Abstract: For each positive integer n, we construct a Steiner triple system of order with no almost parallel class; that is, with no set of disjoint triples. In fact, we construct families of -designs with an analogous property. The only previously known examples of Steiner triple systems of order congruent to without almost parallel classes were the projective triple systems of order with n odd, and 2 of the 11,084,874,829 Steiner triple systems of order 19. [Copyright &y& Elsevier]

Novák conjectured in 1974 that for any cyclic Steiner triple systems of order v with v ≡ 1 (mod 6) , it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic (v , k , λ) -designs with 1 ⩽ λ ⩽ k − 1. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all v , but we conjecture that it holds whenever v is sufficiently large compared to k. We confirm that the generalization of the conjecture holds when v is prime and λ = 1 and also when λ ⩽ (k − 1) / 2 and v is sufficiently large compared to k. As a corollary, we show that for any k ⩾ 3 , with the possible exception of finitely many composite orders v , every cyclic (v , k , 1) -design without short orbits is generated by a (v , k , 1) -disjoint difference family. [ABSTRACT FROM AUTHOR]

Abstract: The block graph of a Steiner triple system of order v is a strongly regular graph. For large v, every strongly regular graph with these parameters is the block graph of a Steiner triple system, but exceptions exist for small orders. An explanation for some of the exceptional graphs is here provided via the concept of switching. (Group divisible designs corresponding to) Latin squares are also treated in an analogous way. Many new strongly regular graphs are obtained by switching and by constructing graphs with prescribed automorphisms. In particular, new strongly regular graphs with the following parameters that do not come from Steiner triple systems or Latin squares are found: , , , , , and . [Copyright &y& Elsevier]

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: 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]

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]

A Steiner triple system (STS(v)) is said to be 3-balanced if every 3-colouring of it is equitable; that is, if the cardinalities of the colour classes differ by at most one. A 3-colouring, φ, of an STS(v) is unique if there is no other way of 3-colouring the STS(v) except possibly by permuting the colours of φ. We show that for every admissible v⩾25, there exists a 3-balanced STS(v) with a unique 3-colouring and an STS(v) which has a unique, non-equitable 3-colouring. [Copyright &y& Elsevier]

In this article we construct uncountably many new homogeneous locally finite Steiner triple systems of countably infinite order as Fraïssé limits of classes of finite Steiner triple systems avoiding certain subsystems. The construction relies on a new embedding result: any finite partial Steiner triple system has an embedding into a finite Steiner triple system that contains no nontrivial proper subsystems that are not subsystems of the original partial system. Fraïssé's construction and its variants are rich sources of examples that are central to model-theoretic classification theory, and recently infinite Steiner systems obtained via Fraïssé-type constructions have received attention from the model theory community. [ABSTRACT FROM AUTHOR]