Pairs of simple balanced ternary designs with prescribed numbers of triples in common

A balanced ternary design of order v with block size three, index 2 and P2 = 1 is a collection of multi-subsets of size 3 (of type { x, y, z} or {x, x, y }) called blocks, in which each unordered pair of distinct elements occurs twice possibly in one block and in which each element is repeated in just one block. So there are precisely v blocks of type {x, x, y}. Necessarily v == 0 (mod 3). We denote such a design by BTD( Vj P2 = 1; 3,2). It is called simple if all its blocks are distinct. Let I1(v) denote the set of numbers k so that there exist two simple BTD(v; P2 = 1; 3, 2) with precisely k blocks in common. (Here the subscript 1 refers to the value of P2.) In this paper we determine the set h (v) for all admissible v == 0 (mod 3). 1 Some preliminaries