Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.23483

Steiner 3-designs as extensions

Michael Kiermaier·Vedran Krčadinac·Alfred Wassermann
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 $46$ and $92$ points. Our construction method is based on extending Steiner $2$-designs using prescribed extension groups. We also consider extensions to designs of higher strength. The article includes a table and a discussion of the status of all admissible parameters for Steiner $3$-designs on at most $50$ points.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael Kiermaier, Vedran Krčadinac, Alfred Wassermann. 2025-09-27. Steiner 3-designs as extensions. https://doi.org/10.1007/s10623-026-01888-w

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On nut graphs with two vertex and three edge orbits

Nut graphs are graphs whose adjacency matrix is singular with one-dimensional null space spanned by a vector with no zero entries. In a recent paper, Bašić, Fowler and Pisanski proved that the automorphism group of a nut graph has more orbits on the edge set than on the vertex set. They classified all orders for which a vertex-transitive nut graph with precisely two edge orbits exists, and conjectured that a nut graph with two vertex and three edge orbits exists for each non-prime order $n \ge 9$. Motivated by this conjecture, we introduce a very general construction that provides graphs with the desired symmetry properties, and we determine some sufficient spectral and structural conditions under which they are nut graphs. The construction yields infinite families of examples and confirms the above conjecture for all odd non-prime orders up to $2\,500$ and for at least $99.8$ percent of all odd non-prime orders up to a million. Finally, we present some additional interesting examples of nut graphs with two vertex and three edge orbits that do not arise from this construction.

math.CO↗
arXiv2026

On vertex-minimal simplicial maps to the sphere

For positive integers $n,d$, let $λ(n,d)$ be the minimal number of vertices of a triangulation of the $n$-sphere which admits a degree $d$ simplicial map onto the boundary of the $(n+1)$-simplex. We show that for $h=\lfloor\frac{n+1}2\rfloor$, the function $λ(n,d)^h$ has linear order of growth in $d$, answering a question of O. Musin. All triangulations we obtained are isomorphic to boundaries of convex polytopes in $\mathbb{R}^{n+1}$.

math.CO↗