Partitions with no Coarsenings of Higher Degree

Abstract

There exist infinite subsets of the natural numbers with no subsets of higher Turing degree. This was established by R. I . SOARE [3] in a construction using the GalvinPrikry theorem. S. G. SIMPSON and T. J. CARLSON [2] proved a "dual" form of the Galvin-Prikry theorem and its extension by E. ELLENTUCK that applies to partitions and their coarsenings rather than to sets and their subsets. SIMPSON conjectured that the "dual" form of SOARE'S result was also true; i.e. that there exist partitions of the natural numbers into infinite numbers of classes with no infinite coarsenings of higher degree. A proof of SIMPSON'S conjecture follows. It employs a construction that mirrors SOARE'S, but uses the dual Ellentuck theorem in place of the Galvin-Prikry theorem.The following notation and terminology are drawn, for the most part, from SIMPSON and CARLSON [a]. Let N denote the natural numbers, including zero, and identify each n E N with the set of smaller natural numbers, i.e. n = (0, . . ., n -l}. If S is either N or some n E N, let a partition of S be a collection of pairwise disjoint, nonempty subsets of S whose union is S. Call the elements of such a partition its blocks, and call the smallest element of each block a leader. If X and Y are partitions of S , say that Y is coarser than X , or that Y i s a coarsening of X , if every block in X is a subset of some block in Y . Call elements of the same block equivalent. Call a partition of N infinite if it contains infinitely many blocks.If X is a partition of N, n E N, and s is a partition of n, say that s i s a segment of X , written s X , if s is the partition of n induced by X , i.e. if s = X [ n ] = { X A n : X E X } -{ O } .If s is a partition of n, call n the length of s, and denote it by lh(s); note that all coarsenings of s have the same length.Let (s, X ) + denote the set of all coarsenings of X , finite or infinite, for which s is a segment.The subsets of (N)" of the form (s, X ) are called dual Ellentuck neighborhoods, and the dual Ellentuck topology on (N)" is the topology whose basic open sets are the dual Ellentuck neighborhoods. The standard topology on (N)" is the topology it inherits l ) This material was part of the author's doctoral dissertation at the Pennsylvania State University [I]. He wishes to thank his dissertation advisor, STEPHEN G. SIMPSON.