On maximum parallel classes in packings

TL;DR

This paper investigates the maximum size of packings with constraints on partial parallel classes for block designs, providing bounds and constructions mainly for the case k=4, with some extensions to larger k.

Contribution

It offers new upper and lower bounds on the maximum number of blocks in packings with a given partial parallel class size, specifically for k=4, along with explicit constructions.

Findings

Derived bounds on β(ρ, v, 4) for various parameters.

Constructed packings close to upper bounds for small ρ.

Extended some methods to cases where k > 4.

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 $k > 4$.