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

Abstract

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.