Extended triple systems having a prescribed number of blocks in common

Abstract

An extended triple system of order υ (ETS (υ)) is a collection of triples of the type {x,y,z}, {x,x,y} or {x,x,x} chosen from a υ-set, such that every pair (not necessarily distinct) belongs to exactly one triple. If ETS(υ) exists with a idempotents, then the number of triples of the system is b_υ,a = (υ(υ + 3) + 2a)/6. In this paper, we show that there are two (not necessarily distinct) ETS(υ)'s with common triples in the set {0, 1, 2,...., b_υ}, where b_υ, is b_υ,υ if υ ≡ 1, 3 (mod 6); b_υ,υ/2 if υ ≡ 0, 2 (mod 6); b_υ,(υ-2)/2 if υ ≡ 4 (mod 6); and b_υ,υ-4 if υ ≡ 5 (mod 6).