Anne Penfold Street

Defining Sets for Block Designs: An Update

This paper deals with the following question: how many, and which, blocks of a design with given parameters must be known before the remaining blocks of the design are uniquely determined? We survey the theoretical background on such defining sets, some specific results for smallest and other minimal defining sets for small designs and the techniques used in finding them, the few known results on minimal defining sets for infinite classes of designs, and the conjectures on minimal defining sets for some classes of Hadamard designs.

Overlarge sets and partial geometries

We extend our earlier work on overlarge sets of Fano planes, obtaining three results of particular interest. We find seven new partial geometries pg(8,7,4) and nine new strongly regular graphs, by means of switching cliques of points with spreads of lines. One of these new strongly regular graphs supports four different partial geometries. Then we give a new construction of the recently discovered eightfold cover of the complete graph K16.

Partitioning sets of quadruples into designs III

All of the non-isomorphic ways of partitioning the collection of all the quadruples chosen from a set of eight elements into five disjoint 2-(8, 4, 3) designs are determined.

DESIGNS WITH PARTIAL NEIGHBOUR BALANCE

A number of authors have written recently on nearest neighbour methods (see Bartlett (1978) and Wilkinson, Eckert, Hancock and Mayo (1983)). In this paper we consider the construction of nearest neighbour designs which satisfy the balance conditions of Wilkinson et al. (1983).

Group ramsey theory

Abstract A subset S of a group G is said to be a sum-free set if S ∩ (S + S) = ⊘ . Such a set is maximal if for every sum-free set T ⊆ G, we have |T| ⩽ |S|. Here, we generalize this concept, defining a sum-free set S to be locally maximal if for every sum free set T such that S ⊆ T ⊆ G, we have S = T. Properties of locally maximal sum-free sets are studied and the sets are determined (up to isomorphism) for groups of small order.

Steiner Trades That Give Rise To Completely Decomposable Latin Interchanges

In this paper we focus on the representation of Steiner trades of volume less than or equal to nine and identify those for which the associated partial latin square can be decomposed into six disjoint latin interchanges.

Construction of Large Sets of Pairwise Disjoint Transitive Triple Systems

A transitive triple is a collection of three ordered pairs of the form {(a, b), (b, c), (a, c)}, where a, b, c are all distinct. A transitive triple system (TTS) of order v is a pair (S, T) where S is a set containing v elements and T is a collection of transitive triples of elements of S such that every ordered pair of distinct elements of S belongs to exactly one transitive triple of T. For all v ≡ 0 or 1 (mod 3), it is well-known that a TTS exists, and that |T| = v(v − 1)/3. Since there are altogether v(v − 1)(v − 2) transitive triples of elements of S, it is natural to ask whether the collection of all transitive triples can be partitioned into 3(v − 2) pairwise disjoint TTSs, or failing that, to find the largest positive integer D(v) for which D(v) pairwise disjoint TTSs of order v exist. We show that D(3v) ⩾ 6v + D(v), and D(3v + 1) ⩾ 6v + D(v + 1), for v ⩾ 2.

The spectrum of minimal defining sets of some Steiner systems

For a design D, define spec(D)={|M||M is a minimal defining set of D} to be the spectrum of minimal defining sets of D. In this note we give bounds on the size of an element in spec(D) when D is a Steiner system. We also show that the spectrum of minimal defining sets of the Steiner triple system given by the points and lines of PG(3,2) equals {16, 17, 18, 19, 20, 21, 22}, and point out some open questions concerning the Steiner triple systems associated with PG(n, 2) in general.

Simple minimum coverings ofK n with copies ofK 4 −e

SummaryAK4−e design of ordern is a pair (S, B), whereB is an edge-disjoint decomposition ofKn (the complete undirected graph onn vertices) with vertex setS, into copies ofK4−e, the graph on four vertices with five edges. It is well-known [1] thatK4−e designs of ordern exist for alln ≡ 0 or 1 (mod 5),n ≥ 6, and that if (S, B) is aK4−e design of ordern then |B| =n(n − 1)/10.Asimple covering ofKn with copies ofK4−e is a pair (S, C) whereS is the vertex set ofKn andC is a collection of edge-disjoint copies ofK4−e which partitionE(Kn)⋃P, for some