Scattered subsets of \(\mathbb{Q}\) (Q1803665)

A linear order is said to be scattered if it has no subset which is order-isomorphic to the set \(\mathbb{Q}\) of rational numbers. If \(A\), \(B\) are scattered subsets of \(\mathbb{Q}\), the author defines \(A\leq B\) if there is some order automorphism \(\varphi\) of \(\mathbb{Q}\) such that \(\varphi(A)\subseteq B\). Then it is proved that the scattered subsets of \(\mathbb{Q}\) form a better quasi-order under the above relation \(\leq\) (embeddability via order automorphisms). This result corresponds to the famous theorem of \textit{R. Laver} [Ann. Math., II. Ser. 93, 89-111 (1971; Zbl 0208.289)] that the scattered orderings form a better quasi-order under embeddability.