Generalized cohesiveness
Generalized cohesiveness
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广义内聚力
DOI：
10.2307/2586482
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发表时间：
1997
影响因子：
0.6
通讯作者：
C. Jockusch
中科院分区：
文献类型：
--
作者：
Tamara Lakins Hummel;C. Jockusch
摘要我们研究了与拉姆齐定理的有效形式有关的自然产生的内聚性的一些广义概念。一个自然数的无限集合A是n-凝聚的（分别是n-r-凝聚的），如果A对每个自然数的n元集合的可计算的（分别是可计算的）2-着色几乎是齐次的。(Thus 1-凝聚集和1-R-凝聚集分别与凝聚集和R-凝聚集一致。我们考虑了n-凝聚集和n-r-凝聚集的不可解度和算术可定义度。例如，我们证明了对所有n ≥ 2，存在一个n-凝聚集。当n = 2时，我们证明了存在一个2-凝聚集，从而改进了这个结果。我们证明了当n ≥ 2时，n-凝聚度和n-r-凝聚度一起构成了一个线性的、非塌缩的度层次。此外，当n ≥ 2时，我们将n-凝聚度的跳刻画为度≥ 0（n+1），并刻画了n-r-凝聚度的跳.
Abstract 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 ≥ 2, there exists a n-cohesive set. We improve this result for n = 2 by showing that there is a 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 ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n-cohesive degrees as exactly the degrees ≥ 0(n+1) and also characterize the jumps of the n-r-cohesive degrees.