Rigid chains admitting many embeddings (Q2706570)

A chain is rigid if it has no non-trivial automorphism and it is superdense if it can be embedded into any of its intervals. Modifying a classical construction of Dushnik and Miller, the authors show that there is a dense subset of \(\mathbb{R}\) which is rigid and superdense. For \(\kappa\) regular and uncountable, they give a completely different construction of a rigid chain of cardinality \(\kappa\). This is done using stationary sets for coding points of a dense subset of the chain under construction. Using finite sequences of stationary sets they construct a rigid superdense chain of cardinality \(\kappa\). They show that their method can be modified to obtain a rigid superdense chain of cardinality \(\kappa\) for any uncountable \(\kappa\).