On maximum parallel classes in packings

Abstract

The integer $\beta (\rho ,v,k)$ is defined to be the maximum number of blocks in any $(v,k)$-packing in which the maximum partial parallel class (or PPC) has size $\rho$. This problem was introduced and studied by Stinson for the case $k=3$. Here, we mainly consider the case $k=4$ and we obtain some upper bounds and lower bounds on $\beta (\rho ,v,4)$. We also provide some explicit constructions of $(v,4)$-packings having a maximum PPC of a given size $\rho$. For small values of $\rho$, the number of blocks of the constructed packings are very close to the upper bounds on $\beta (\rho ,v,4)$. Some of our methods are extended to the cases $k>4$.