Block-avoiding sequencings of points in Steiner triple systems

Good point sequencings of Steiner triple systems

An l-good sequencing of a Steiner triple system of order v, STS(v), is a permutation of the points of the system such that no l consecutive points in the permutation contains a block. It is known that every STS(v) with v > 3 has a 3-good sequencing. It is proved that every STS(v) with v >= 13 has a 4-good sequencing and every 3-chromatic STS(v) with v >= 15 has a 5-good sequencing. Computational results for Steiner triple systems of small order are also given.

Sequencing Partial Steiner Triple Systems

A partial Steiner triple system of order n is sequenceable if there is a sequence of length n of its distinct points such that no proper segment of the sequence is a union of point-disjoint blocks. We prove that if a partial Steiner triple system has at most three point-disjoint blocks, then it is sequenceable.

Block-avoiding point sequencings of arbitrary length in Steiner triple systems

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 $\ell$-good sequencing of any STS$(v)$ provided that $v$ is sufficiently large. We also prove some new nonexistence results for $\ell$-good sequencings of STS$(v)$.

Countable homogeneous Steiner triple systems avoiding specified subsystems

In this article we construct uncountably many new homogeneous locally finite Steiner triple systems of countably infinite order as Fra\"ıssé limits of classes of finite Steiner triple systems avoiding certain subsystems. The construction relies on a new embedding result: any finite partial Steiner triple system has an embedding into a finite Steiner triple system that contains no nontrivial proper subsystems that are not subsystems of the original partial system. Fra\"ıssé's construction and its variants are rich sources of examples that are central to model-theoretic classification theory, an…

部分 Steiner 系统的避块点序列化

Block avoiding point sequencings of partial Steiner systems

本文证明了当 $n$ 变大时，任何部分 $(n,k,t)_λ$-系统都存在 $\ell=Θ(n^{1/t})$ 的 $\ell$-好的序列化，改进了已有结果，并给出了部分 Steiner 三元系统的特例。

Block-avoiding point sequencings of Mendelsohn triple systems

A cyclic ordering of the points in a Mendelsohn triple system of order $v$ (or MTS$(v)$) is called a sequencing. A sequencing $D$ is $\ell$-good if there does not exist a triple $(x,y,z)$ in the MTS$(v)$ such that (1) the three points $x,y,$ and $z$ occur (cyclically) in that order in $D$; and (2) $\{x,y,z\}$ is a subset of $\ell$ cyclically consecutive points of $D$. In this paper, we prove some upper bounds on $\ell$ for MTS$(v)$ having $\ell$-good sequencings and we prove that any MTS$(v)$ with $v \geq 7$ has a $3$-good sequencing. We also determine the optimal sequencings of every MTS$(v)$…