Some Results on Transverse Steiner Quadruple Systems of Type $g^t{u}^1$

Abstract

A transverse Steiner quadruple system $(TSQS)$ is a triple $(X,\mathcal{H},\mathcal{B})$ where $X$ is a $v$-element set of points, $\mathcal{H}=\{H_1,H_2,\dots ,H_r\}$ is a partition of $X$ into holes, and $\mathcal{B}$ is a collection of transverse $4$-element subsets with respect to $\mathcal{H}$, called blocks, such that every transverse $3$-element subset is in exactly one block. In this article, we study transverse Steiner quadruple systems with $r$ holes of size $g$ and $1$ hole of size $u$. Constructions based on the use of $s$-fans are given, including a construction for quadrupling the number of holes of size $g$. New results on systems with $6$ and $11$ holes are obtained, and constructions for $\text{TSQS}(x^n(2n{)}^1)$ and $\text{TSQS}(4^n{2}^1)$ are provided.