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Tamara Hummel [6]Tamara J. Hummel [1]Tamara Lakins Hummel [1]
  1.  28
    Generalized cohesiveness.Tamara Hummel & Carl G. Jockusch - 1999 - Journal of Symbolic Logic 64 (2):489-516.
    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 (...)
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  2.  13
    ∑2-constructions and I∑1.Marcia Groszek & Tamara Hummel - 1998 - Annals of Pure and Applied Logic 93 (1-3):83-101.
    The consistency strength of the ∑2 priority method is I∑2, yet classical theorems proven by this method have been proved from I∑1. Is there a statement about the structure of the r.e. degrees that can be proved using a ∑2 argument and cannot be proved from I∑1?We rule out statements in the language of partial orderings of the form …[], where is quantifier-free, by showing that the following can be proved in I∑1.If P is any recursive partial ordering with a (...)
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    ∑2-constructions and I∑1.Marcia Groszek & Tamara Hummel - 1998 - Annals of Pure and Applied Logic 93 (1-3):83-101.
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    ∑2-constructions and I∑1.Marcia Groszek & Tamara Hummel - 1998 - Annals of Pure and Applied Logic 93 (1-3):83-101.
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  5.  34
    Ramsey's theorem for computably enumerable colorings.Tamara J. Hummel & Carl G. Jockusch - 2001 - Journal of Symbolic Logic 66 (2):873-880.
    It is shown that for each computably enumerable set P of n-element subsets of ω there is an infinite Π 0 n set $A \subseteq \omega$ such that either all n-element subsets of A are in P or no n-element subsets of A are in P. An analogous result is obtained with the requirement that A be Π 0 n replaced by the requirement that the jump of A be computable from 0 (n) . These results are best possible in (...)
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  6.  9
    Effective Versions of Ramsey's Theorem: Avoiding the Cone Above $\mathbf{0}$'.Tamara Lakins Hummel - 1994 - Journal of Symbolic Logic 59 (4):1301-1325.
    Ramsey's Theorem states that if $P$ is a partition of $\lbrack\omega\rbrack^\kappa$ into finitely many partition classes, then there exists an infinite set of natural numbers which is homogeneous for $P$. We consider the degrees of unsolvability and arithmetical definability properties of infinite homogeneous sets for recursive partitions. We give Jockusch's proof of Seetapun's recent theorem that for all recursive partitions of $\lbrack\omega\rbrack^2$ into finitely many pieces, there exists an infinite homogeneous set $A$ such that $\emptyset' \nleq_T A$. Two technical extensions (...)
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