Direct, easy, and self-contained proofs are presented of the existence of triple systems with A = 2 having no repeated blocks.
1. Ilhoduction
A A-fold triple system S,(2,3, u) is a way of selecting unordered triples from a u-set so that any given pair of elements lies in precisely A triples. An S, (2,3, u) is simple if it contains no repeated triple.
An &(2,3, u) can exist only if o = 0 or 1 (mod 3). Bose [l] showed in 1939 that these conditions are sufficient. More recently, Van Buggenhaut [6] has proven that there is a simple Z&(2,3, u) in every such case (except, obviously, 2, = 3). However, his construction assumes knowledge of Hanani's inductive construction [3] of 3-wise balanced designs with block sizes {4,6} of all orders, and of Doyen's construction [2] of disjoint Steiner triple systems of all orders v = 3 (mod 6). He also uses pairwise balanced designs with block sizes (3,s) and other designs, and constructs eight specific cases. Street [S] has produced an interesting construction in terms of other triple systems, but again it is recursive.
Our aim here is to give direct constructions of simple S,(2,3, v) for all possible v, based only on the idea of a Latin square. (No use of the properties of Latin squares is made, other than those defined herein.)
The longest part of the paper is the construction of what we call 'skew transversal squares' for orders congruent to 3 and 4 modulo 6; and these are only used or needed for S,(2,3, v) with v = 10 or 13 (mod 18).
2. !SomeLatinsqnarea
A Latin square of order n is an n x n array whose rows and columns are all permutations of an n-set. Unless otherwise stated, our Latin squares are based on