Q-degrees of n-C.E. sets

In this paper we study Q-degrees of n-computably enumerable (n-c.e.) sets. It is proved that n-c.e. sets form a true hierarchy in terms of Q-degrees, and that for any n ≥ 1 there exists a 2n-c.e. Q- degree which bounds no noncomputable c.e. Q-degree, but any (2n + l)- c.e. non 2n-c.e. Q-degree bounds a c.e. noncomputable Q-degree. Studying weak density properties of n-c.e. Q-degrees, we prove that for any n ≥ 1, properly n-c.e. Q-degrees are dense in the ordering of c.e. Q-degrees, but there exist c.e. sets A and B such that A - B <Q A ≡Q φ′, and there are no c.e. sets for which the Q-degrees are strongly between A - B and A. ©2007 University of Illinois

Random strings and tt-degrees of Turing complete C.E. sets

We investigate the truth-table degrees of (co-)c.e.\ sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefix-free machine is Turing complete, but that truth-table completeness depends on the choice of universal machine. We show that for such sets of random strings, any finite set of their truth-table degrees do not meet to the degree~0, even within the c.e. truth-table degrees, but when taking the meet over all such truth-table degrees, the infinite meet is indeed~0. The latter result proves a conjecture of Allender, Friedman and Gasarch. We also show that there are two Turing complete c.e. sets whose truth-table degrees form a minimal pair.Comment: 25 page

Generalized cohesiveness

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set $A$ of natural numbers is $n$--cohesive (respectively, $n$--r--cohesive) if $A$ is almost homogeneous for every computably enumerable (respectively, computable) $2$--coloring of the $n$--element sets of natural numbers. (Thus the $1$--cohesive and $1$--r--cohesive sets coincide with the cohesive and r--cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of $n$--cohesive and $n$--r--cohesive sets. For example, we show that for all $n \ge 2$, there exists a $\Delta^0_{n+1}$ $n$--cohesive set. We improve this result for $n = 2$ by showing that there is a $\Pi^0_2$ $2$--cohesive set. We show that the $n$--cohesive and $n$--r--cohesive degrees together form a linear, non--collapsing hierarchy of degrees for $n \geq 2$. In addition, for $n \geq 2$ we characterize the jumps of $n$--cohesive degrees as exactly the degrees {\bf \geq \jump{0}{(n+1)}} and show that each $n$--r--cohesive degree has jump {\bf > \jump{0}{(n)}}