For each admissible v, we exhibit a path design P (v, 4, 1) with a spanning set of minimum cardinality and a P (v, 4, 1) with a scattering set of maximum

In this paper, we remove the outstanding values m for which the existence of a GBTD(4,m) has not been decided previously. This leads to a complete solution to the existence problem regarding GBTD(4,m)s.

The existence of (M, B)-optimal designs for block size three is completely settled. Such a design is a collection of b triples on v elements so that for every two elements the numbers of triples containing them differ by at most one, and for every two pairs of elements the numbers of triples containing them also differ by at most one.

B) is said to have the Size Four property if the blocks in B can be partitioned into two isomorphic sets. In this paper we construct block with four with the property, thus the exception of 6 A. Rosa [4]).

The minimum number of blocks having maximum size precisely four that are required to cover, exactly λ times, all pairs of elements from a set of cardinality v is denoted by g λ (v) (or g (4)(v) when λ = 1). All values of g (4) λ (v) are known except for λ = 1 and v = 17 or 18. It is known that 30 ≤ g(4)(17) ≤ 31 and 32 ≤ g(4)(18) ≤ 33. In this paper we show that g(4)(17) 6= 30 and g(4)(18) 6= 3...

The design (15,21,7,5,2) is the only BIBD of block size 5 that does not exist. If it did exist, it would provide an exact bicovering, in 21 blocks of size 5, of the pairs from 15 points. However, we show that 22 quintuples are sufficient to provide a bicover of the pairs from 15 points; thus there are only 10 repetitions required in the bicovering.

The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: λ(v − 1) ≡ 0 mod (k − 1) λv(v − 1) ≡ 0 mod k(k − 1) Earlier work has studied k = 9 with λ ∈ {1, 2, 3, 4, 6, 8, 9, 12}. In this article we show that the necessary conditions are sufficient for λ = 9 and every other λ not previously studied.

The purpose of this paper is to present a large number of highly A-efficient incomplete block designs for making comparisons among a set of test treatments and a control treatment. These designs are BTIB designs. A simple method of construction of BTIB designs, based on BIB designs is proposed. The advantage of this method is that one can use the vast literature on BIB designs to obtain a large...

The existence of modiied group divisible designs with block size four is settled with a handful of possible exceptions.