A class of extended triple systems and numbers of common blocks - Huang, Wen-Chung (Utilitas Mathematica, 2000)
An extended triple system of order v with one idempotent element (ETS(v, 1)) is a collection of triples of the type {x, y, z}, {x, x, y} or {x, x, x} chosen from a v-set, such that every pair (not necessarily distinct) belongs to exactly one triple and there is only one triple of the type {x, x, x}. A necessary and sufficient condition for the existence of such a design is v ≢ 0 (mod 3). In this paper, we have constructed two ETS(v,1)'s such that the number of common triples is in the following set: (1) {0, 1, 2, . . . , bv - 3, bv - 2, bv}, for even v ≥ 4, and (2) {0, 1, 2, . . . , bv - 4, bv - 3, bv}, for odd v ≥ 11, where bv = (v + 2)(v + 1)/6.