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arXiv · 2007.11033

On partial parallel classes in partial Steiner triple systems

Douglas R. Stinson
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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.

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Douglas R. Stinson. 2020-07-21. On partial parallel classes in partial Steiner triple systems. https://arxiv.org/abs/2007.11033

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