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Semiselective coideals

  • Ilijas Farah (a1)

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In this note we give an answer to the following problem of Todorcevic: Find out the combinatorial essence behind the fact that the family ℋ of the ground-model infinite sets of integers in a Perfect-set forcing extension has the property that for any Borel f: [ℕ]ω → {0, 1} there exists an A ∈ ℋ such that f is constant on [A]ω (see [7], [13]). In other words, one needs to capture the combinatorial properties of the family ℋ of ground-model subsets of ℕ which assure that it diagonalizes all Borel partitions. It turns out that the notion which results from our analysis of this problem is a bit more optimal than the older notion of a “happy family” (or selective coideal) introduced by A.R.D. Mathias [16] long ago in order to extend the well-known theorems of Galvin–Prikry [6] and Silver [25] (see Theorems 3.1 and 4.1 below). We should remark that these Mathias-style extensions can indeed be as useful in the applications as the original partition theorems.

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Semiselective coideals

  • Ilijas Farah (a1)

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