A bound for Wilson's theorem (III)

✍ Scribed by Yanxun Chang

• Publisher: John Wiley and Sons
• Year: 1996
• Tongue: English
• Weight: 486 KB
• Volume: 4
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/(sici)1520-6610(1996)4:2<83::aid-jcd1>3.0.co;2-v

✦ Synopsis

In this article we prove the following statement. For any positive integers k 2 3 and A, let &A) =exp{exp{k'*}}. If Av(v -1) = 0 (mod k(k -I)) and A(v -1) = 0 (mod k -1) and v > c ( k , A), then a B(v, k , A) exists. o 1996 John Wiley & Sons, Inc.

1. Introduction

A painvise balanced design (or PBD) is a pair (X, A ) , where A is a collection of subsets (called blocks) of X , each of cardinality at least two, such that every unordered pair of points (i.e., elements of X ) is contained in exactly A blocks in A (we allow so-called "repeated blocks"). If v is a positive integer and K is a set of positive integers, each of which is greater than or equal to 2, then we say that ( X , A ) is a ( v , K , A ) -PBD if 1x1 = v and (A1 E K for every A E A . We define B ( K , A ) = {v : there exists a (v, K , A) -PBD}, and abbreviate B ( K , I ) by B ( K ) .

We write B ( k , A) for the set of all v such that a B ( v , k , A) exists, and we write B(k, 1) briefly as B ( k

and That is, the congruences (1.1) and (1.2) are necessary conditions for the existence of a B ( v , k , A ) .