A local Ramsey theory for block sequences

Smythe, Iian

doi:10.1090/tran/7448

A local Ramsey theory for block sequences
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by Iian B. Smythe PDF

Trans. Amer. Math. Soc. 370 (2018), 8859-8893 Request permission

Abstract:

We develop local forms of Ramsey-theoretic dichotomies for block sequences in infinite-dimensional vector spaces, analogous to Mathias’s selective coideal form of Silver’s theorem for analytic partitions of $[\mathbb {N}]^\infty$. Under large cardinals, these results are extended to partitions in $\mathbf {L}(\mathbb {R}),$ and $\mathbf {L}(\mathbb {R})$-generic filters of block sequences are characterized. Variants of these results are also established for block sequences in Banach spaces and for projections in the Calkin algebra.

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