2.1 Notation from model theory................... 4 2.2 F

this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25

Key words Degree spectra of algebraic structures, immune and hyperimmune sets and degrees

Abstract. Fraïssé studied countable structures S through analysis of the age of S, i.e., the set of all finitely generated substructures of S. We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal.

Abstract. We study the connection between the number of ascending and descending cuts of a linear order, its classical size, and its effective complexity (how much [how little] information can be encoded into it). 1.