  We continue the investigation of systems of hereditarily rigid relations
started in Couceiro, Haddad, Pouzet and Sch\"olzel [1]. We observe that on a
set $V$ with $m$ elements, there is a hereditarily rigid set $\mathcal R$ made
of $n$ tournaments if and only if $m(m-1)\leq 2^n$. We ask if the same
inequality holds when the tournaments are replaced by linear orders. This
problem has an equivalent formulation in terms of separation of linear orders.
Let $h_{\rm Lin}(m)$ be the least cardinal $n$ such that there is a family
$\mathcal R$ of $n$ linear orders on an $m$-element set $V$ such that any two
distinct ordered pairs of distinct elements of $V$ are separated by some member
of $\mathcal R$, then $ \lceil \log_2 (m(m-1))\rceil\leq h_{\rm Lin}(m)$ with
equality if $m\leq 7$. We ask whether the equality holds for every $m$. We
prove that $h_{\rm Lin}(m+1)\leq h_{\rm Lin}(m)+1$. If $V$ is infinite, we show
that $h_{\rm Lin}(m)= \aleph_0$ for $m\leq 2^{\aleph_0}$. More generally, we
prove that the two equalities $h_{\rm Lin}(m)= log_2 (m)= d({\rm Lin}(V))$
hold, where $\log_2 (m)$ is the least cardinal $\mu$ such that $m\leq 2^\mu$,
and $d({\rm Lin}(V))$ is the topological density of the set ${\rm Lin}(V)$ of
linear orders on $V$ (viewed as a subset of the power set $\mathcal{P}(V\times
V)$ equipped with the product topology). These equalities follow from the {\it
Generalized Continuum Hypothesis}, but we do not know whether they hold without
any set theoretical hypothesis.
