Related papers: Embedding partial Steiner triple systems with few …

We show that a pair of orthogonal partial latin squares of order $n$ can be embedded in a pair of orthogonal latin squares of order at most $16n^4$ and all orders greater than or equal to $48n^4$. This paper provides the first direct…

We try to find all quadruples of positive integers $(m,a,b,c)$ with $a \geq b \geq c$ such that there exists a distinct covering system with minimum modulus $m$ and least common multiple of the moduli $2^a 3^b 5^c$. We obtain complete…

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 Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…

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 $46$…

Several natural partial orders on integral partitions, such as the embeddability, the stable embeddability, the bulk embeddability and the supermajorization, raise in the quantum computation, bin-packing and matrix analysis. We find the…

Spatial models of preference, in the form of vector embeddings, are learned by many deep learning and multiagent systems, including recommender systems. Often these models are assumed to approximate a Euclidean structure, where an…

We report on numerical studies of two-dimensional electron systems in the presence of a perpendicular magnetic field, with a high Landau level index (N > 1) half filled by electrons. Strong and sharp peaks are found in the wave vector…

In this paper new Steiner systems $S(2,6,121)$, $S(2,6,126)$, $S(2,7,169)$ are introduced. Also some non-existence results for line lengths $7..11$ are presented. There is no solid proof that presented algorithm is exhaustive or correct,…

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…

For given integers $r$ and $\ell$ such that $2\leqslant\ell\leqslant r-1$, an $r$-uniform hypergraph $H$ is called a partial Steiner $(n,r,\ell)$-system, if every subset of size $\ell$ lies in at most one edge of $H$. In particular, partial…

We enumerate total cyclic orders on $\left\{1,\ldots,n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(i,{i+1},{i+2})$, these integers being taken modulo $n$. In some cases, the problem reduces to the…

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…

In this paper, we consider outlier embeddings into HSTs. In particular, for metric $(X,d)$, let $k$ be the size of the smallest subset of $X$ such that all but that subset (the ``outlier set'') can be probabilistically embedded into the…

Metric embeddings traditionally study how to map $n$ items to a target metric space such that distance lengths are not heavily distorted; but what if we only care to preserve the relative order of the distances (and not their length)? In…

To investigate objects without a describable notion of distance, one can gather ordinal information by asking triplet comparisons of the form "Is object $x$ closer to $y$ or is $x$ closer to $z$?" In order to learn from such data, the…

We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure $\mathcal A$ as the family of Turing degrees that compute…

In this note six Steiner systems $S(2,8,225)$ and four Steiner systems $S(2,9,289)$ are presented. This resolves two of $129$ undecided cases for block designs with block length $8$ and $9$, mentioned in Handbook of Combinatorial Designs.

In this paper new $1$-rotational 2-Steiner systems for different admissible $v,k$ pairs are introduced. In particular, $1$-rotational unitals of order $4$ are enumerated.

We present a criterion of local Normal Embedding of a semialgebraic (or definable in an o-minimal structure) contained in $R^n$ in terms orders of contact of arcs. Namely, we prove that a semialgebraic set is normally embedded at a point x…