The Number of Repeated Blocks in Two Fold Extended Triple Systems

Abstract

A twofold extended triple system with two idempotent elements, \(TETS(v)\), is a pair \((V, B)\), where \(V\) is a \(v\)-set and \(B\) is a collection of triples, called blocks, of type \(\{x,y,z\}\), \(\{x,x,y\}\) or \(\{x,x,x\}\) such that every pair of elements of \(V\), not necessarily distinct, belongs to exactly two triples and there are only two triples of the type \(\{x, x, x\}\).
This paper shows that an indecomposable \(TETS(v)\) exists which contains exactly \(k\) pairs of repeated blocks if and only if \(v \not\equiv 0 \mod 3\), \(v \geq 5\) and \(0 \leq k \leq b_v – 2\), where \(b_v = \frac{(v + 2)(v + 1)}{6}\).