Related papers: Mixed Steiner Triples Systems with Shortest Length

A mixed Steiner system MS$(t,k,Q)$ is a set (code) $C$ of words of weight $k$ over an alphabet $Q$, where not all coordinates of a word have the same alphabet size, each word of weight $t$, over $Q$, has distance $k-t$ from exactly one…

A famous theorem of Kirkman says that there exists a Steiner triple system of order $n$ if and only if $n\equiv 1,3\mod{6}$. In 1973, Erd\H{o}s conjectured that one can find so-called `sparse' Steiner triple systems. Roughly speaking, the…

For a numerical monoid $\langle n_1, \dots, n_k \rangle$ minimally generated by $n_1, \dots, n_k \in \mathbb{N}$ with $n_1 < \cdots < n_k$, the longest and shortest factorization lengths of an element $x$, denoted as $L(x)$ and $\ell(x)$,…

Let $G$ be a simple finite graph and $G'$ be a subgraph of $G$. A $G'$-design $(X,\cal B)$ of order $n$ is said to be embedded into a $G$-design $(X\cup U,\cal C)$ of order $n+u$, if there is an injective function $f:\cal B\rightarrow \cal…

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…

In 1847, Kirkman proved that there exists a Steiner triple system on $n$ vertices (equivalently a triangle decomposition of the edges of $K_n$) whenever $n$ satisfies the necessary divisibility conditions (namely $n\equiv 1,3 \mod 6$). In…

This paper considers two closely related concepts, mixed Steiner system and nonuniform group divisible design (GDD). The distinction between the two concepts is the minimum Hamming distance, which is required for mixed Steiner systems but…

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…

We prove that with high probability $\mathbb{G}^{(3)}(n,n^{-1+o(1)})$ contains a spanning Steiner triple system for $n\equiv 1,3\pmod{6}$, establishing the exponent for the threshold probability for existence of a Steiner triple system. We…

We establish an upper bound on the minimum codegree necessary for the existence of spanning, fractional Steiner triple systems in $3$-uniform hypergraphs. This improves upon a result by Lee in 2023. In particular, together with results from…

We prove several structural properties of Steiner triple systems (STS) of order 3w+3 that include one or more transversal subdesigns TD(3,w). Using an exhaustive search, we find that there are 2004720 isomorphism classes of STS(21)…

The existence of Steiner triple systems STS(n) of order n containing no nontrivial subsystem is well known for every admissible n. We generalize this result in two ways. First we define the expander property of 3-uniform hypergraphs and…

We study the existence over small fields of Maximum Distance Separable (MDS) codes with generator matrices having specified supports (i.e. having specified locations of zero entries). This problem unifies and simplifies the problems posed…

Let $D(n)$ be the number of pairwise disjoint Steiner quadruple systems. A simple counting argument shows that $D(n) \leq n-3$ and a set of $n-3$ such systems is called a large set. No nontrivial large set was constructed yet, although it…

We construct Steiner triple systems without parallel classes for an infinite number of orders congruent to $3 \pmod{6}$. The only previously known examples have order $15$ or $21$.

Recently, generalizations of the classical Three Gap Theorem to higher dimensions attracted a lot of attention. In particular, upper bounds for the number of nearest neighbor distances have been established for the Euclidean and the maximum…

In a graph, $k$ cycles are {\em admissible} if their lengths form an arithmetic progression with common difference one or two. Let $G$ be a 2-connected graph with minimum degree at least $k\geqslant 4$. We prove that \begin{itemize} \item…

We prove that for all natural numbers $m$ and $k$ where $k$ is odd, there exists a natural number $N(k)$ such that any 3-connected cubic graph with at least $N(k)$ vertices contains a cycle of length $m$ modulo $k$. We also construct a…

An $\ell$-good sequencing of an STS$(v)$ is a permutation of the points of the design such that no $\ell$ consecutive points in this permutation contain a block of the design. We prove that, for every integer $\ell \geq 3$, there is an…

A partial Steiner triple system is is $sequenceable$ if the points can be sequenced so that no proper segment can be partitioned into blocks. We show that, if $0 \leq a \leq (n-1)/3$, then there exists a nonsequenceable PSTS$(n)$ of size…