We investigate the existence of large sets of t-designs. We introduce t-wise equivalenceand (n, t)-partitionable sets. We propose a general approach to construct largesets of t-designs. Then, we consider large sets of a prescribed size n. We partitionthe set of all k-subsets of a v-set into several parts, each can be written as productof two trivial designs. Utilizing these partitions we develop some recursive methodsto construct large sets of t-designs. Then, we direct our attention to the large setsof prime size. We prove two extension theorems for these large sets. These theoremsare the only known recursive constructions for large sets which do not put anyadditional restriction on the parameters, and work for all t and k. One of them,has even a further advantage; it increase the strength of the large set by one, and itcan be used recursively which makes it one of a kind. Then applying this theoremrecursively, we construct large sets of t-designs for all t and some blocksizes k.
Hartman conjectured that the necessary conditions for the existence of a largeset of size two are also sufficient. We suggest a recursive approach to the Hartmanconjecture, which reduces this conjecture to the case that the blocksize is a powerof two, and the order is very small. Utilizing this approach, we prove the Hartmanconjecture for t = 2. For t = 3, we prove that this conjecture is true for infinitelymany k, and for the rest of them there are at most k/2 exceptions.
In Chapter 4 we consider the case k = t + 1. We modify the recursive methodsdeveloped by Teirlinck, and then we construct some new infinite families of largesets of t-designs (for all t), some of them are the smallest known large sets. We alsoprove that if k = t + 1, then the Hartman conjecture is asymptotically correct.
