3 research outputs found
On large sets of disjoint steiner triple systems III
AbstractTo construct large sets of disjoint STS(3n) (i.e., LTS(3n)), we introduce a new kind of combinatorial designs. Let S be a set of n elements. If x β S, we denote an n Γ n square array on S by Ax, if for every w β S\{x} the following conditions are satisfied: Ax = [ayz(x)](y, z β S), axx(x) = x, aww(x) β w, axw(x) = axw(x) = x, and {awz(x) | z β S} = {ayw(x) | y β S} = S. Let j β {1, 2&}, Aj = {ayz[j](y, z β S) be a Latin square of order n based on S with n parallel transversals including the diagonal one. Two squares Ax and Axβ² on the same S are called disjoint, if ayz(x) β ayz(xβ²) whenever y, z β S\{x, xβ²}; two squares Ax and Aj on the same S are called disjoint, if ayz(x) β ayz[j] whenever y, z β S\ {x}; and two squares A1 and A2 on the same S are called disjoint, if ayz[1] β ayz[2] whenever y β ez. It is a set of n + 2 pairwise disjoint squares Ax (x runs over S), A1 and A2 on S as mentioned above that is very useful to construct LTS(3n), and such a set we denote by LDS(n). The essence in the relation between LDS(n) and LTS(3n) is the following theorem which is established in the Section 2:Theorem. If there exist both an LDS(n) and an LTS(n + 2), then there exists an LTS(3n) also.The set of integers n for which LDS(n) exist is denoted by D. In the other parts of this paper, the following results are given: 1.(1) If n β D, and q = 2Ξ± (Ξ± is an integer greater than 1), or q β {;5, 7, 11, 19}, then qn β D.2.(2) If pΞ± is a prime power, p > 2 and pΞ± β D, then 3pΞ± β D.3.(3) If q is a prime power greater than 4 and 1 + n β D, then 1 + qn β D.4.(4) If t is a nonnegative integer, then 7 + 12t β D and 5 + 8t β D
Large Sets of t-Designs
We investigate the existence of large sets of t-designs. We introduce t-wise equivalence
and (n, t)-partitionable sets. We propose a general approach to construct large
sets of t-designs. Then, we consider large sets of a prescribed size n. We partition
the set of all k-subsets of a v-set into several parts, each can be written as product
of two trivial designs. Utilizing these partitions we develop some recursive methods
to construct large sets of t-designs. Then, we direct our attention to the large sets
of prime size. We prove two extension theorems for these large sets. These theorems
are the only known recursive constructions for large sets which do not put any
additional 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 it
can be used recursively which makes it one of a kind. Then applying this theorem
recursively, 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 large
set of size two are also sufficient. We suggest a recursive approach to the Hartman
conjecture, which reduces this conjecture to the case that the blocksize is a power
of two, and the order is very small. Utilizing this approach, we prove the Hartman
conjecture for t = 2. For t = 3, we prove that this conjecture is true for infinitely
many 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 methods
developed by Teirlinck, and then we construct some new infinite families of large
sets of t-designs (for all t), some of them are the smallest known large sets. We also
prove that if k = t + 1, then the Hartman conjecture is asymptotically correct.</p