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 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.