On partial parallel classes in partial Steiner triple systems

Abstract

For an integer $\rho$ such that $1\le \rho \le v/3$, 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.