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)}}

-Generic Computability, Turing Reducibility and Asymptotic Density

Generic computability has been studied in group theory and we now study it in the context of classical computability theory. A set A of natural numbers is generically computable if there is a partial computable function f whose domain has density 1 and which agrees with the characteristic function of A on its domain. A set A is coarsely computable if there is a computable set C such that the symmetric difference of A and C has density 0. We prove that there is a c.e. set which is generically computable but not coarsely computable and vice versa. We show that every nonzero Turing degree contains a set which is not coarsely computable. We prove that there is a c.e. set of density 1 which has no computable subset of density 1. As a corollary, there is a generically computable set A such that no generic algorithm for A has computable domain. We define a general notion of generic reducibility in the spirt of Turing reducibility and show that there is a natural order-preserving embedding of the Turing degrees into the generic degrees which is not surjective

Asymptotic density and the Ershov hierarchy

We classify the asymptotic densities of the $\Delta^0_2$ sets according to their level in the Ershov hierarchy. In particular, it is shown that for $n \geq 2$, a real $r \in [0,1]$ is the density of an $n$-c.e.\ set if and only if it is a difference of left-$\Pi_2^0$ reals. Further, we show that the densities of the $\omega$-c.e.\ sets coincide with the densities of the $\Delta^0_2$ sets, and there are $\omega$-c.e.\ sets whose density is not the density of an $n$-c.e. set for any $n \in \omega$.Comment: To appear in Mathematical Logic Quarterl

Reducibilities in recursive function theory.

Massachusetts Institute of Technology. Dept. of Mathematics. Thesis. 1966. Ph.D.Bibliography: leaves 102-103.Ph.D

On a triangulation of the 3-ball and the solid torus

AbstractWe show that neither the 3-ball nor the solid torus admits a triangulation in which (i) every vertex is on the boundary, and (ii) every tetrahedron has exactly one triangle on the boundary. Such triangulations are relevant to an unresolved conjecture of Perles