Recent results show that a constraint satisfaction problem (CSP) defined over rational numbers with their natural ordering has a solution if and only if it has a definable solution. The proof uses advanced results from topology and modern model theory. The aim of this paper is threefold. (1) We give a simple purely-logical proof of the claim and show that the advanced results from topology and model theory are not needed; (2) we introduce an intrinsic characterisation of the statement "definable CSP has a solution iff it has a definable solution" and investigate it in general intuitionistic set theories (3) we show that the results from modern model theory are indeed needed, but for the implication reversed: we prove that "definable CSP has a solution iff it has a definable solution" holds over a countable structure if and only if the automorphism group of the structure is extremely amenable.

中文翻译：

---

约束满足问题的顺应性

最近的结果表明，在具有自然排序的有理数上定义的约束满足问题 (CSP) 当且仅当它具有可定义的解决方案时才有解决方案。该证明使用拓扑和现代模型理论的高级结果。本文的目的有三个。(1) 我们给出了一个简单的纯逻辑证明，并表明不需要拓扑和模型理论的高级结果；(2) 我们引入了“可定义的 CSP 有一个解决方案，当它有一个可定义的解决方案”这一陈述的内在特征，并在一般直觉集理论中对其进行研究 (3) 我们表明现代模型理论的结果确实是需要的，但是对于含义颠倒了：我们证明“可定义的 CSP 有一个解决方案，如果它有一个可定义的解决方案”