Related papers: Simple signed Steiner triple systems

Stern's diatomic sequence is a well-studied and simply defined sequence with many fascinating characteristics. The binary signed-digit representation of integers is an alternative representation of integers with much use in efficient…

In 2000, Rees and Shalaby constructed simple indecomposable two-fold cyclic triple systems for all v congruent to 0, 1, 3, 4, 7, and 9 (mod 12) where v = 4 or v>11, using Skolem-type sequences. We construct, using Skolem-type sequences,…

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)…

Cycle switching is a particular form of transformation applied to isomorphism classes of a Steiner triple system of a given order $v$ (an $STS(v)$), yielding another $STS(v)$. This relationship may be represented by an undirected graph. An…

In a recent work, Jungnickel, Magliveras, Tonchev, and Wassermann derived an overexponential lower bound on the number of nonisomorphic resolvable Steiner triple systems (STS) of order $v$, where $v=3^k$, and $3$-rank $v-k$. We develop an…

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…

An avoidance problem of configurations in 4-cycle systems is investigated by generalizing the notion of sparseness, which is originally from Erd\H{o}s' r-sparse conjecture on Steiner triple systems. A 4-cycle system of order v, 4CS(v), is…

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 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…

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 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…

Stern's diatomic sequence is a well-studied and simply defined sequence with many fascinating characteristics. The binary signed-digit (BSD) representation of integers is used widely in efficient computation, coding theory and other…

A $\delta$-colouring of the point set of a block design is said to be {\em weak} if no block is monochromatic. The {\em chromatic number} $\chi(S)$ of a block design $S$ is the smallest integer $\delta$ such that $S$ has a weak…

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 $21$.

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…

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.…

We introduce a uniform method of proof for the following results. For {\em each} of the following conditions, there are $2^{\aleph_0}$ families of Steiner systems, satisfying that condition: i) Theorem~2.2.4: (extending \cite{Chicoetal})…

Suppose that P is an infinite set of primes such that P = A + B + C, where A,B,C are sets with at least two elements. We show that if P(x) > c x/log^d x (where P(x) = the number of elements of P that are <= x), and if A,B,C is a "regular"…

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 -…

An $r$-block-coloring, simply $r$-coloring, of a Steiner triple system $\mathrm{STS}(v)$ is a partition of the block set into $r$ color classes, each color class being a partial parallel class. The chromatic index of $\mathrm{STS}(v)$,…