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 $v\not\equiv 0\mathrm{mod}3$, $v\ge 5$ and $0\le k\le b_v–2$, where $b_v=\frac{(v+2)(v+1)}{6}$.