The spectrum of λ-times repeated blocks for TS(υ,λ)

✍ Scribed by Salvatore Milici; Zsolt Tuza

• Book ID: 103058994
• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 494 KB
• Volume: 129
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(92)00510-x

✦ Synopsis

A triple system TS(u, A) is a collection of 3-element subsets (= blocks) of a v-element set X such that each pair x, yeX is contained in precisely 1 blocks. We determine the spectrum of i-times repeated blocks in a TS(r, i), leaving only one value in doubt for each v = 2 (mod 12).

1. Introduction

A triple system of order u and index 1, TS(u, A), is a pair ( V, B), where V is a u-element set and B is a collection of 3-element subsets of V, called triples or blocks, such that each 2-element subset of V is contained in exactly 2 blocks. This definition permits repeated blocks. It is well-known [6] that a TS(u,A) exists if and only if L(u--I)-0(mod2)

and Au(u-l)=O(mod6). The support of a triple system ( V, B) is the set B* E B of distinct blocks; the support size is the number of distinct blocks, IB*l.

Recently, there has been substantial interest [l-5,8,9,11,12] to determine (i) the spectrum of possible support sizes of a TS( u, A), (ii) the spectrum of repeated blocks in a TS(u, 2).

The determination of possible support sizes for TS(u, A) with 2 < 8 is essentially completed [l, 2,4,12]. The spectrum of repeated blocks has been examined for 1, = 3 C3,