We then apply this theorem recursively and construct large sets of t-designs for all t and some block sizes k. We adapt the recursive methods developed by Teirlinck, and then we construct some new infinite families of large sets of t-designs (for all £), some of which are the smallest known large sets. Finally, we give a brief overview of the known results on the existence of large sets.
If 6 = c = d = 0, then this theorem can be applied recursively to obtain an infinite family of large groups. To the author's knowledge Theorems 2, 3 and 4 are the only known recursive constructions for large groups which do not impose any additional restrictions on pa. This can be determined by applying methods similar to those used to construct large arrays of the first size along with some simple counting arguments.
We follow Teirlinck's approach to find some new large sets of t-designs (for all t), some of which are of lower order than those found in [41]. In order to apply our construction recursively, we restrict ourselves to (f t* l)-trivial large sets of t-designs.
Preliminaries
The Incidence M atrix
The assertion is an immediate consequence of the identity 1.3 and the fact that a t-trade is also an i-trade for i < t. Then the necessary conditions for the existence of a 5(A; t, k, v) can be written, the compact form is A(£, k, u)|A. In this section, we examine the well-known results for the existence of large sets of t-designs.
It is known that the necessary conditions for the existence of an A-factorization of Kn are also sufficient. In light of the above comments, our approach to constructing large sets is as follows. We end this section with the following lemma, which is a very simple yet useful application of t-wise equivalent sets to the construction of large sets.
Therefore, given the above remarks, we must prove the claim that we must show that. Then 5 has a (Z-trivial) partition into I mutually t-blue equivalent subsets if and only if IF has a (Z-trivial) partition into I mutually t-blue equivalent subsets. Now it is time to look again at equivalence classes, so that we can find w such that a given equivalence class has a (t-trivial) partition into I mutually t-blue equivalent subsets.
It is clear that r(Xli„.iimj is of the desired form. t -r 1 — m) exists, then C has a Z-trivial partition into I mutually t-wise equivalent subsets. Assume, for S € P0,t-i(K) there exists an integer bs such that bs divides gi(S) for all i, and the equivalence class T{S) has a ((t -f- l)-trivial ) partition into as = a/bs t-wise equivalent subsets. In this section, we apply Theorems 3.2 to obtain some results on the existence of large sets of size 2.
Teirlinck, A completion of Lu's determination of the spectrum for large sets of Steiner triple systems, J.
The Structure of (t, k, t/)-Ttades
Review of Known Results
On the other hand, the necessary conditions are also sufficient for any other value of v. The existence of this large set, together with some well-known recursive constructions, establishes the Hartman conjecture for t = 6, k = 7. Then, with the derivative and the residues of the large sets in this family, we can confirm the Hartman conjecture for k < 7 .
In particular, he proved that for a given t there exists A(t) such that the necessary conditions for ex. In this chapter, we introduce the notion of t-wise equivalent sets, which is essentially a generalization of (t, k, v) deals. In the first section we will discuss their importance in constructing large arrays and find some recursive procedures for constructing them, and in the second section we will use t-wise equivalent arrays to find several recursive constructs.
Finally in the third part, to emphasize even more their importance and strength, we show how this simple idea can lead to the construction of large groups of first size and small order for all t. Before continuing along this line, we would like to make a few remarks about the above definition to clarify its importance in the construction of large groups. Bn} be a partition of Pk(X) into n mutually equivalent t-th sets. Now, the numbers of occurrences of each T of Pt{X) in all B i's are the same, and in the next generation, every subset t of X in total appears in exactly blocks of B i's.
Bm are mutually disjoint subgroups of Pk(X) such that each of them has a partition into n mutually t-equivalent sets, then their union, U&& = B also has a partition into n t-wise equivalent sets. First, we present some procedures to construct the (n, t) disjoint sets (sets that have a partition into n equivalent disjoint subsets from point "t") horn the (n, ti) disjoint sets (ti < t ) and then we give a partition of Pk{X) into such groups. The following lemma shows how (n, t)-separating groups can be constructed from older ones. ii) if Bi has a partition into m reciprocally equivalent subsets t2, then B\ * Bi has a partition into m reciprocally equivalent subgroups (£i •+• t2 -+- l)-si.
It is clear that each Bi is of the desired shape. We will show that they form a partition Pk{X).
Large Sets of Prime S iz e
We define a function
In fact, we show that for infinitely many values of k the Hartman conjecture is true. It should be noted that the methods we use for t = 3 can be used in general to obtain some partial results for t > 3. To do this, it will be more convenient to express the necessary conditions as a system of congruence relations .
The Case t = 2
The proof of the following lemma is straightforward and therefore does not appear here. In the next section, by means of a series of Lemmas, we prove that if n > 5, and the statement is true for n — 1, then it is also true for n which completes the proof. As we discussed in Chapter 1, S(X;t,k,v) designs are equivalent to the nonnegative integer solutions of the following nonhomogeneous system of linear equations. M v ) F = AI, (4.1).
Note that this method can reduce the size of the original system of equations by a factor of |G| at most and for a given group G there is no guarantee that S(A; t, k, v) is designed by G as an automorphism group exists (In fact, there are many t-plans with a trivial automorphism group.), so for large values of v this method is not practical . At this point it may seem that we have reduced the problem of the existence of t-plans to the more difficult problem of the existence of f-plans with a prescribed group of automorphisms. Therefore, we can replace agi(S)l§ with aSi in the definition of Bi s. If we do this for all S 6 A.t+iOO, then Bi s will be o-uniform and disconnected.
In this section we follow Teirlinck's approach to find some new large sets of t-designs (for all t), some of which have smaller order than those constructed in Theorem 4.1.5. The main idea is to work with (t -r l)-trivial large sets of t-designs, allowing us to replace (t - Irregularity (in Teirlinck's construction) with a much stronger condition of 1-regularity. Two subsets of are called equivalent if they have the same cardinality and the same support.
