o-minimality (Q1129865)

A structure \({\mathcal M} =(M,<, \dots)\), where \(<\) is a dense linear order (without endpoints) of the domain \(M\), is called o-minimal if every definable (without parameters) subset of \(M\) is a finite union of points and open intervals (with endpoints in \(M\cup \{\pm \infty \})\). Tarski proved that if \(\mathbb R\) is the real numbers, then the ordered ring \((\mathbb R; +,\cdot,0, 1,<)\) is o-minimal. Let \(\overline{\mathbb R}\) be any expansion of the real ordered field \(\mathbb R\) with a language \(\overline L\). Call a formula \(\psi\) of \(\overline L\) tame if there exists a natural number \(N\) (depending only on \(\psi)\) such that whenever the free variables of \(\psi\) are partitioned into two classes \[ \psi= \psi(x_1, \dots, x_m,y_1, \dots, y_r), \] then the set \[ \bigl\{ \langle b_1,\dots, b_r\rangle \in\mathbb R^r: \overline{\mathbb R}\models\psi [a_1, \dots, a_n,b_1, \dots, b_r] \bigr\} \] has at most \(N\) connected components for any choice of \(\langle a_1, \dots, a_n \rangle \in\mathbb R^n\). In this paper the author asks for counterexamples to the following conjecture: With \(\overline{\mathbb R}\) as above, if every quantifier-free formula of \(L\) is tame, then is \(\mathbb R\) o-minimal? -- Which, in fact, would imply that every formula of \(\overline L\) is tame.