Related papers: Enumerations of maximum partial triple systems on …
A Steiner triple system is a set $S$ together with a collection $\mathcal{B}$ of subsets of $S$ of size 3 such that any two elements of $S$ belong to exactly one element of $\mathcal{B}$. It is well known that the class of finite Steiner…
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…
Steiner systems are a fascinating topic of combinatorics. The most studied Steiner systems are $S(2, 3, v)$ (Steiner triple systems), $S(3, 4, v)$ (Steiner quadruple systems), and $S(2, 4, v)$. There are a few infinite families of Steiner…
A partial Steiner triple system of order $u$ is a pair $(U,\mathcal{A})$ where $U$ is a set of $u$ elements and $\mathcal{A}$ is a set of triples of elements of $U$ such that any two elements of $U$ occur together in at most one triple. If…
In this article, we construct a Steiner system with the parameters $S(3,6,42)$, settling one of the smallest open parameter sets of Steiner $3$-designs. Furthermore, we establish the existence of rotational Steiner quadruple systems on $46$…
In this paper, we initiate the study of discrepancy questions for combinatorial designs. Specifically, we show that, for every fixed $r\ge 3$ and $n\equiv 1,3 \pmod{6}$, any $r$-colouring of the triples on $[n]$ admits a Steiner triple…
In 1973 Erdos asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g \ge 4 for which some g-element vertex-set contains at…
Computational techniques for the construction of quasi-symmetric block designs are explored and applied to the case with $56$ points. One new $(56,16,18)$ and many new $(56,16,6)$ designs are discovered, and non-existence of $(56,12,9)$ and…
A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 was recently carried out in [P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The perfect binary one-error-correcting…
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…
The intersection of two Steiner triple systems (X,A) and (X,B) is the set A intersect B. The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m,n) such that…
In this article we present the largest set of unitals (totally 553) in projective planes of order 16. An open question is what is the number of the known unitals that are non-isomorphic to the reported ones. The results are obtained with a…
The 2-block intersection graph (2-BIG) of a twofold triple system (TTS) is the graph whose vertex set is composed of the blocks of the TTS and two vertices are joined by an edge if the corresponding blocks intersect in exactly two elements.…
In this article we construct uncountably many new homogeneous locally finite Steiner triple systems of countably infinite order as Fra\"{\i}ss\'{e} limits of classes of finite Steiner triple systems avoiding certain subsystems. The…
We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a…
We initiate the study of extended bicolorings of Steiner triple systems (STS) which start with a $k$-bicoloring of an STS($v$) and end up with a $k$-bicoloring of an STS($2v+1$) obtained by a doubling construction, using only the original…
Let $n$, $k$, and $t$ be integers satisfying $n>k>t\ge2$. A Steiner system with parameters $t$, $k$, and $n$ is a $k$-uniform hypergraph on $n$ vertices in which every set of $t$ distinct vertices is contained in exactly one edge. An…
We show that for any n divisible by 3, almost all order-n Steiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almost resolvable).
In this article we study extensions of Steiner triple systems by means of the associated Steiner loops. We recognize that the set of Veblen points of a Steiner triple system corresponds to the center of the Steiner loop. We investigate…
A directed triple system of order $v$ (or, DTS$(v)$) is decomposition of the complete directed graph $\vec{K_v}$ into transitive triples. A $v$-good sequencing of a DTS$(v)$ is a permutation of the points of the design, say $[x_1 \; \cdots…