Malcolm Greig

Existence of (v; {5 ;w ∗ }; 1)-PBDs

Abstract In this paper, we investigate the existence of pairwise balanced designs on v points having blocks of size five, with a distinguished block of size w, briefly (v,{5,w ∗ },1) -PBDs. The necessary conditions for the existence of a (v,{5,w ∗ },1) -PBD with a distinguished block of size w are that v⩾4w+1, v≡w≡1 ( mod 4) and either v≡w ( mod 20) or v+w≡6 ( mod 20) . Previously, w⩽33 has been studied, and the necessary conditions are known to be sufficient for w=1, 5, 13 and 21, with 8 possible exceptions when w⩽33. In this article, we eliminate 3 of these possible exceptions, showing sufficiency for w=25 and 33. Our main objective is the study of 37⩽w⩽97, where we establish sufficiency for w=73, 81, 85 and 93, with 67 possible exceptions with 37⩽w⩽97. For w≡13 ( mod 20) , we show that the necessary existence conditions are sufficient except possibly for w=53,133,293 and 453. For w≡1,5 ( mod 20) , we show the necessary existence conditions are sufficient for w⩾1281,1505, and for w≡9,17 ( mod 20) , we show that w⩾2029,2477 is sufficient with one possible exceptional series, namely v=4w+9 when w≡17 ( mod 20) . We know of no example where v=4w+9. In this article, we also study the 4-RBIBD embedding problem for small subdesigns (up to 52 points) and update some results of Bennett et al. on PBDs containing a 5-line. As an application of our results for w=33 and 97, we establish the smallest number of blocks in a pair covering design with k=5 when v≡1 ( mod 4) with 37 open cases, the largest being for v=489; hitherto, there were 104 open cases, the largest being v=2249.

Balanced Incomplete Block Designs with Block Size 7

The object of this paper is the construction of balanced incomplete block designs with k=7. This paper continues the work begun by Hanani, who solved the construction problem for designs with a block size of 7, and with λ=6, 7, 21 and 42. The construction problem is solved here for designs with λ > 2 except for v=253, λ= 4,5 ; also for λ= 2, the number of unconstructed designs is reduced to 9 (1 nonexistent, 8 unknown).

Resolvable Balanced Incomplete Block Designs with BlockSize 8

The necessary condition for the existence of a resolvable balanced incomplete block design on v points, with λ = 1 and k = 8, is that v ≡ 8 mod 56. With the exception of 66 values of v, this condition is shown to be sufficient. The largest exceptional value of v is 24480.

Resolvable balanced incomplete block designs with block size 5

Abstract A necessary condition for the existence of a resolvable balanced incomplete block design on v points with λ=2 and k=5, is that v≡5 mod 10 . This condition is shown to be sufficient for v>395, with at most 13 exceptions below this value. Also given are a new (185,5,1) RBIBD, (v,5,4) RBIBDs for v=110 and 140, and (45,5,λ) RBIBDs for λ⩾3.

Generalized whist tournament designs

In this study a new class of tournament designs is introduced. In particular, each game of the tournament involves several (two or more) teams competing against one another. The tournament is also required to satisfy certain balance conditions that are imposed on each pair of players. These balance conditions are related to both the total number of players on each team and the number of teams in each game. In one sense, these balance conditions represent a generalization of the balance requirements for whist tournaments although the games in a whist tournament involve, exclusively, two two-player teams. Several techniques for constructing these new tournament designs are developed and theorems guaranteeing infinite classes of such designs are proven.

Pair covering designs with block size 5

In this article we look at pair covering designs with a block size of 5 and v=0(mod4). The number of blocks in a minimum covering design is known as the covering number C(v,5,2). For v= =28 with v=0(mod4), it seems probable that C(v,5,2)=L(v,5,2). We establish this for all but 17 possible exceptional values lying in the range 40=

Balanced incomplete block designs with block size 9: part II

Abstract 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) . In this paper we study k=9 with λ=3, 6 and 12. We show that these conditions on v are sufficient, with the possible exceptions of v=177, 345, 385 when λ=3, and v=213 when λ=6. We give constructions of TD3(10,n)s with 13 possible exceptions, namely n∈{5,6,14,20,35,45,55,56,60,78,84,85,102}; we also reduce the number of unknown TDs with block sizes 8 and 9.

Maximal arcs in Steiner systems S(2,4,v)

A maximal arc in a Steiner system S(2,4,v) is a set of elements which intersects every block in either two or zero elements. It is well known that v=4(mod12) is a necessary condition for an S(2,4,v) to possess a maximal arc. We describe methods of constructing an S(2,4,v) with a maximal arc, and settle the longstanding sufficiency question in a strong way. We show that for any v=4(mod12), we can construct a resolvable S(2,4,v) containing a triple of maximal arcs, all mutually intersecting in a common point. An application to the motivating colouring problem is presented.

Balanced Incomplete Block Designs with Block Size 9 and λe2,4,8

AbstractThe necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: