A local Ramsey theory for block sequences

Abstract

We develop local forms of Ramsey-theoretic dichotomies for block sequences in infinite-dimensional vector spaces, analogous to Mathias' 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.

08/20/2018: Posted final pre-publication version. To appear in Trans. Amer. Math. Soc

References in corpus (1)

• Local Ramsey theory. An abstract approach

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