In joint work with Ekaterina Fokina and Luca San Mauro we investigate a related notion, bi-embeddability spectra. Two structures are bi-embeddable if each is embeddable in the other and the bi-embeddability spectrum of a structure is the set of Turing degrees of its bi-embeddable copies. We give general results on bi-embeddability spectra and investigate bi-embeddability spectra of strongly locally finite graphs.
In 2014, M. Cai, Lempp, J. Miller and Soskova discovered an unusual structural property of the continuous degrees: if we join a continuous degree with a total degree that is not below it then the result is always a total degree. We call degrees with this curious property almost total. We prove that the almost total degrees coincide with the continuous degrees. Since the total degrees are definable in the partial order of the enumeration degrees, this implies that the continuous degrees are also definable. Applying earlier work of J. Miller on the continuous degrees, this shows that the relation "PA above" on the total degrees is definable in the enumeration degrees.
In order to prove that every almost total degree is continuous, we pass through another characterization of the continuous degrees that slightly simplifies one of Kihara and Pauly. Like them, we identify our almost total degree as the degree of a point in a computably regular space with a computable dense sequence, and then we apply the effective version of Urysohn's metrization theorem to reveal our space as a computable metric space.
This is joint work with Uri Andrews, Greg Igusa, and Joseph Miller.
In joint work with Harrison-Trainor, we use these ideas to define a class of structures being universal within finitely generated structures. We then use small cancelation techniques from group theory to show that the class of finitely generated groups with finitely many constants named is universal within finitely generated structures.
I will discuss some results on the computability theory and reverse mathematics of theorems concerning basic model-theoretic notions such as atomic, homogeneous, and saturated models, including connections with combinatorial principles, and with first-order principles such as forms of induction restricted to formulas of low complexity. I will draw in particular from my recent paper "Induction, bounding, weak combinatorial principles, and the Homogeneous Model Theorem" with Karen Lange and Richard Shore.