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Partitioning Subsets of Stable Models

Timothy Bays
The Journal of Symbolic Logic
Vol. 66, No. 4 (Dec., 2001), pp. 1899-1908
DOI: 10.2307/2694983
Stable URL: http://www.jstor.org/stable/2694983
Page Count: 10
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Partitioning Subsets of Stable Models
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Abstract

This paper discusses two combinatorial problems in stability theory. First we prove a partition result for subsets of stable models: for any A and B, we can partition A into |B|$^{< \kappa(T)}$ pieces, $\langle$ Ai ∣ i < |B|$^{< \kappa(T)}\rangle$, such that for each Ai there is a B$_i \subseteq$ B where |Bi| < κ(T) and Ai &2ADD; $\underset{B_i}$ B. Second, if A and B are as above and |A| > |B|, then we try to find A' $\subset$ A and B' $\subset$ B such that |A'| is as large as possible, |B'| is as small as possible, and A' &2ADD; $\underset{B'}$ B. We prove some positive results in this direction, and we discuss the optimality of these results under ZFC + GCH.

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