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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]

This work firstly studies the Steiner tree problem with bounded edge-length d in which d is the ratio of the maximum edge cost to the minimum edge cost. This work analyzes the algorithm of Byrka et al. [19] and shows that the approximation ratio of d ln ⁡ 4 d + ln ⁡ 4 − 1 + ϵ for general graphs and approximation ratio of 73 ⋅ d 60 ⋅ d + 13 + ϵ for quasi-bipartite graphs. The algorithm implies approximation ratio of 1.162 + ϵ for the problem on complete graphs with edge distances 1 and 2. This finding represents an improvement upon the previous best approximation ratio of 1.25. This work then presents a combinatorial two-phase heuristic for the general Steiner tree in greedy strategy that achieves an approximation ratio of 1.4295. [ABSTRACT FROM AUTHOR]

Pseudo-geometric designs are combinatorial designs which share the same parameters as a finite geometry design, but are not isomorphic to that design. As far as we know, many pseudo-geometric designs have been constructed by the methods of finite geometries and combinatorics. However, none of pseudo-geometric designs with the parameters S (2 , q + 1 , (q n − 1) / (q − 1)) is constructed by the approach of coding theory. In this paper, we use cyclic codes to construct pseudo-geometric designs. We firstly present a family of ternary cyclic codes from the m -sequences with Welch-type decimation d = 2 ⋅ 3 (n − 1) / 2 + 1 , and obtain some infinite family of 2-designs and a family of Steiner systems S (2 , 4 , (3 n − 1) / 2) using these cyclic codes and their duals. Moreover, the parameters of these cyclic codes and their shortened codes are also determined. Some of those ternary codes are optimal or almost optimal. Finally, we show that one of these obtained Steiner systems is inequivalent to the point-line design of the projective space PG (n − 1 , 3) and thus is a pseudo-geometric design. [ABSTRACT FROM AUTHOR]

If S is a transitive metric space, then | C | ⋅ | A | ≤ | S | for any distance- d code C and a set A , "anticode", of diameter less than d. For every Steiner S (t , k , n) system S , we show the existence of a q -ary constant-weight code C of length n , weight k (or n − k), and distance d = 2 k − t + 1 (respectively, d = n − t + 1) and an anticode A of diameter d − 1 such that the pair (C , A) attains the code–anticode bound and the supports of the codewords of C are the blocks of S (respectively, the complements of the blocks of S). We study the problem of estimating the minimum value of q for which such a code exists, and find that minimum for small values of t. [ABSTRACT FROM AUTHOR]

The research on orthogonal Steiner systems S (t , k , v) was initiated in 1968. For (t , k) ∈ { (2 , 3) , (3 , 4) } , this corresponds to orthogonal Steiner triple systems (STSs) and Steiner quadruple systems (SQSs), respectively. The existence problem of a pair of orthogonal STSs or SQSs was settled completely thirty years ago. However, for Steiner systems with t ≥ 3 and k ≥ 5 , only two small examples of orthogonal pairs were known to exist before this work. An infinite family of orthogonal Steiner systems S (3 , 5 , v) is constructed in this paper. In particular, the existence of a pair of orthogonal Steiner systems S (3 , 5 , 4 m + 1) is established for any even m ≥ 2 ; additionally a pair of orthogonal G-designs G ( 4 m + 1 5 , 5 , 5 , 3) is displayed for any odd m ≥ 3. The construction is based on the Steiner systems admitting 3-transitive automorphism groups supported by elementary symmetric polynomials. Moreover, 50 mutually orthogonal Steiner systems S (5 , 8 , 24) are shown to exist. [ABSTRACT FROM AUTHOR]

Inspired by an L p Steiner formula for the L p affine surface area proved by Tatarko and Werner, we define, in analogy to the classical Steiner formula, L p -Steiner quermassintegrals. Special cases include the classical mixed volumes, the dual mixed volumes, the L p affine surface areas and the mixed L p affine surface areas. We investigate the properties of the L p -Steiner quermassintegrals in a special class of convex bodies. In particular, we show that they are rotation and reflection invariant valuations in this class of convex bodies with a certain degree of homogeneity. Such valuations seem new and have not been observed before. [ABSTRACT FROM AUTHOR]

We study reflection positivity in the context of Hilbert space, and Krein-space theory. Our context is that of triple systems (U , J , M +) assumed to satisfy the axioms for reflection positivity (also known as Osterwalder-Schrader positivity). Applications include quantum field theory and the theory of unitary representations of Lie groups. Our analysis and classification make use of Krein space theory, and of realizations of input/output systems. We present a new and explicit classification of the case when triples (U , J , M +) constitute reflection positive systems. A key point in our analysis is a specific choice of signature operator J , and Krein space, leading then to an operator theoretic and geometric classification formula for reflection positive triples (U , J , M +) , with U assumed unitary and J -self-adjoint, and M + reflecting the choices of the associated J -positive and U -invariant subspaces. We further consider the wider setting when U is only assumed J -self-adjoint, and we demonstrate how choices of U may be characterized by specific partially defined and contractive operators E. Then the corresponding admissible subspaces M + will be realized as spaces G (X) , graph of a second linear operator X , where the possibilities for such X , as operators, are decided by a non-linear equation (in X), called the Riccati equation. Our present applications include non-commutative harmonic analysis. [ABSTRACT FROM AUTHOR]

The Turán number of hypergraphs has been studied extensively. Here we deal with a recent direction, the linear Turán number, and restrict ourselves to linear triple systems, a collection of triples on a set of points in which any two triples intersect in at most one point. For a fixed linear triple system F , the linear Turán number ex L (n , F) is the maximum number of triples in a linear triple system with n points that does not contain F as a subsystem. We initiate the study of the linear Turán number for an acyclic F. In this case ex L (n , F) is linear in n and we aim for good bounds. Since the case of trees is already difficult for graphs (Erdős–Sós conjecture), we concentrate on matchings, paths and small trees. In case of matchings, where M k is the set of k pairwise disjoint triples, we prove that for fixed k and large enough n , ex L (n , M k) = f (n , k) where f (n , k) is the maximum number of triples that can meet k − 1 points in a linear triple system on n points. This is an analogue of an old result of Erdős on hypergraph matchings. For the k -edge linear path P k we show (extending some standard path increasing methods used for graphs) that ex L (n , P k) ≤ 1. 5 k n which is probably far from best possible. Finding ex L (n , F) relates to difficult problems on Steiner triple systems and interesting even for small trees. For example, for P 4 , the path with four triples, ex L (n , P 4) ≤ 4 n 3 with equality only for disjoint union of affine planes of order 3. On the other hand, for E 4 , the tree having three pairwise disjoint triples and a fourth one meeting all of them, we have bounds only: 6 ⌊ n − 3 4 ⌋ ≤ ex L (n , E 4) ≤ 2 n. [ABSTRACT FROM AUTHOR]

This paper concentrates on mutually disjoint 2-chromatic Steiner quadruple systems of order v (briefly by SQS(v)s), meaning that the SQS(v)s are mutually disjoint and they are mutually 2-chromatic having the same two color classes. Following Phelps and Rosa's work, we study maximum sets of mutually disjoint 2-chromatic SQS(v)s by applying direct constructions for suitable collections of triple representatives of V q by utilizing automorphism groups of affine transformations in the finite fields F q for prime powers q. Our main result presents many new maximum sets of mutually disjoint 2-chromatic SQS(v)s. We also improve the lower bounds on the number of mutually disjoint SQS(v)s as a by-product. [ABSTRACT FROM AUTHOR]

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]

Let K be an abelian group of order v. A Steiner quadruple system of order v (SQS (v)) (K , B) is called symmetric K -invariant if for each B ∈ B , it holds that B + x ∈ B for each x ∈ K and B = − B + y for some y ∈ K. When the Sylow 2-subgroup of K is cyclic, a necessary and sufficient condition for the existence of a symmetric K -invariant SQS (v) was given by Munemasa and Sawa, which is a generalization of a necessary and sufficient condition for the existence of a symmetric cyclic SQS (v) shown in Piotrowski's thesis in 1985. In this paper, we prove that a symmetric K -invariant SQS (v) exists if and only if v ≡ 2 , 4 (mod 6) , the order of each element of K is not divisible by 8, and there exists a symmetric cyclic SQS (2 p) for any odd prime divisor p of v. [ABSTRACT FROM AUTHOR]