Hereditary rigidity, separation and density: In memory of Professor I.G. Rosenberg

We continue the investigation of systems of hereditarily rigid relations started in Couceiro, Haddad, Pouzet and Schölzel [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
$h_{\text{Lin}}(m)$
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
$[\log_{2}(m(m-1))]\leq h_{\text{Lin}}(m)$
with equality if
$m\leq 7$
. We ask whether the equality holds for every
$m$
. We prove that
$h_{\text{Lin}}(m+1)\leq h_{\text{Lin}}(m)+1$
. If
$V$
is infinite, we show that
$h_{\text{Lin}}(m)=\aleph_{0}$
for
$m\leq 2^{\aleph_{0}}$
. More generally, we prove that the two equalities
$h_{\text{Lin}}(m)=log_{2}(m)=d$
(Lin
$(V)$
) hold, where
$\log_{2}(m)$
is the least cardinal
$\mu$
such that
$m\leq 2^{\mu}$
, and
$d$
(Lin
$(V)$
) is the topological density of the set 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 Generalized Continuum Hypothesis, but we do not know whether they hold without any set theoretical hypothesis.