# Block orderings for twofold triple systems

A $\lambda$-fold triple system of order $v$ consists of a $v$-set $V$ and

a collection of 3-subsets (called blocks or triples) of $V$ such that each

2-subset of $V$ occurs in exactly $\lambda$ of the system's triples.

Given a $\lambda$-fold triple system with $\lambda > 1$, we can ask

whether its triples can be ordered so that the union of any two

consecutive triples consists of four elements of $V$. We will describe

some potential applications, briefly review previous results, and discuss

some recent work concerning the nonexistence / existence of such

orderings, with emphasis on 2-fold triple systems. This is joint work

with Aras Erzurumluo\u{g}lu.