Fig. 1.

A collection $\tau =\{\{a,b,c\},\{a,b,d\},\{b,c,e\},\{d,e,f\}\}$ of four sets that is phylogenetically flexible and (ii) a collection ${\tau }^{\prime }=\tau \cup \{\{b,d,e\}\}$ that fails to be phylogenetically flexible. In (i) all of the $3^4=81$ choices of rooted triples (one for each of the four leaf sets) give a set of rooted triples that is displayed by at least one rooted phylogenetic tree on the six leaves $a,b,\dots ,f$. However, in (ii) this fails, for example, the set of rooted triples ab|c, bd|a, bc|e, df|e together with be|d (for the fifth set) is not displayed by any tree on the six leaves. The set $\tau$ is thin, but ${\tau }^{\prime }$ is not, since it has a subset (namely ${\tau }^{\prime }$ itself) which has strictly negative excess (equal to $-1$)