Related papers: $3$-pyramidal Steiner Triple Systems
A $3$-$(v,\{4,6\},1)$ design is a configuration of $v$ points and a collection of $4$- and $6$-element subsets called blocks, that jointly contain every 3-element subset exactly once. Using an exhaustive computer search on $v\leq 28$ points…
We determine all posible orders of automorphisms of finite order of complex K3 surfaces or of K3 surfaces in characteristic $p>3$. E.g., a positive integer $N$ is the order of an automorphism of a complex K3 surface if and only if…
Let S denote a Steiner triple system on an n-element set. An orientation of S is an assignment of a cyclic ordering to each of the triples in S. From an oriented Steiner triple system, one can define an anticommutative bilinear operation on…
Whereas Steiner systems $S(2,k,v)$ with block length $k \le 5$ have large amount of examples and the existence is established for all admissible $v$, for $k\ge 6$ only few examples are known even for decided cases. In this paper the…
We address several extremal problems concerning the spreading property of point sets of Steiner triple systems. This property is closely related to the structure of subsystems, as a set is spreading if and only if there is no proper…
In this paper, we study the problem of finding the largest possible set of s points and s blocks in a Steiner triple system of order v, such that that none of the s points lie on any of the s blocks. We prove that s \leq (2v+5 -…
The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System $S(t,n,v)$ we associate two ideals, in a suitable polynomial ring, defining a Steiner…
In this paper we introduce enumeration of unitals of order $5$, which are also Steiner systems $S(2,6,126)$, where automorphism group acts transitively and effectively on points or fixes one point.
We determine all possible orders of automorphisms of complex K3 surfaces. A positive integer N is the order of an automorphism of a complex K3 surface if and only if $\phi(N) \leq 20$ where $\phi$ is the Euler function.
Given an integer $k\ge3$ and a group $G$ of odd order, if there exists a $2$-$(v,k,1)$-design and if $v$ is sufficiently large, then there is such a design whose automorphism group has a subgroup isomorphic to $G$. A weaker result is proved…
For $m \geq 3$, we define the $m$th order pyramidal number by \[ \mathrm{Pyr}_m(x) = \frac{1}{6} x(x+1)((m-2)x+5-m). \] In a previous paper, written by the first-, second-, and fourth-named authors, all solutions to the equation…
We commence the study of domination in the incidence graphs of combinatorial designs. Let $D$ be a combinatorial design and denote by $\gamma(D)$ the domination number of the incidence (Levy) graph of $D$. We obtain a number of results…
In this paper various Steiner systems $S(2,k,v)$ for $k = 6$ are collected and enumerated for specific constructions. In particular, two earlier unknown types of $1$-rotational designs are found for the groups $SL(2,5)$ and $((\mathbb Z_3…
In this paper we firstly review how to \textit{explicitly} solve a system of $3$ \textit{first-order linear recursions }and outline the main properties of these solutions. Next, via a change of variables, we identify a class of systems of…
Gilbert--Steiner problem is a generalization of the Steiner tree problem on a specific optimal mass transportation. We show that every branching point in a solution of the planar Gilbert--Steiner problem has degree 3.
For an integer $\rho$ such that $1 \leq \rho \leq v/3$, define $\beta(\rho,v)$ to be the maximum number of blocks in any partial Steiner triple system on $v$ points in which the maximum partial parallel class has size $\rho$. We obtain…
A configuration of points and lines is cyclic if it has an automorphism which permutes its points in a full cycle. A closed formula is derived for the number of non-isomorphic connected cyclic configurations of type (v_3), i.e., which have…
We establish the existence of simple designs with parameters $2$-$(55,10,4)$, $3$-$(20,5,4)$, $3$-$(21,7,30)$, $4$-$(15,5,2)$, $4$-$(16,8,45)$, $5$-$(16,7,10)$, and $5$-$(17,8,40)$, which have previously been unknown. For the corresponding…
Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper $3$-system $(P_{1},P_{2},P_{3})$ with the property $\overline{\varphi}_{3}=1$. In fact, our method generalizes to provide $n$-systems with…
We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and $3 \times 3$ matrices. Firstly, we completely…