On partial parallel classes in partial Steiner triple systems

TL;DR

This paper investigates the maximum size of partial Steiner triple systems with constraints on the largest partial parallel class, providing bounds and constructions that reveal their growth rates and applications.

Contribution

It establishes bounds on the maximum number of blocks in partial Steiner triple systems with a given partial parallel class size, including explicit constructions and asymptotic analysis.

Findings

eta( ho,v) is linear in v when ho is constant.

eta( ho,v) is quadratic in v when ho scales with v.

Bounds on eta( ho,v) differ by a constant for fixed ho.

Abstract

For an integer $\rho$ such that $1 \leq \rho \leq v/3$, define $\beta(\rho,v)$ to be the maximum number of blocks in any partial Steiner triple system on $v$ points in which the maximum partial parallel class has size $\rho$. We obtain lower bounds on $\beta(\rho,v)$ by giving explicit constructions, and upper bounds on $\beta(\rho,v)$ result from counting arguments. We show that $\beta(\rho,v) \in \Theta (v)$ if $\rho$ is a constant, and $\beta(\rho,v) \in \Theta (v^2)$ if $\rho = v/c$, where $c$ is a constant. When $\rho$ is a constant, our upper and lower bounds on $\beta(\rho,v)$ differ by a constant that depends on $\rho$. Finally, we apply our results on $\beta(\rho,v)$ to obtain infinite classes of sequenceable partial Steiner triple systems.