A class of extended triple systems and numbers of common triples

An extended triple system with no idempotent element (ETS) is a collection of non ordered triples of type {x,y,z} or {x,x,y} chosen from a v set in such a way that each pair (whether distinct or not) is contained in exactly one triple. (For example, in the block {x,x,y}, the pair {x,y} is said to occur one tinle.) Such a design has "'v = v(v + 3) /6 blocks and a necessary and sufficient condition for existence is that v ° (mod 3). Let J( v) denote the set of non negative integers k such that there exist two ETS(v) with precisely k blocks in common. In this paper we determine J( v) for all admissible v, in particular we show that J(9) 1(9)-{13} and J(v) =1(v), where l{v) ={O,l, ... , sv-3, s~}