  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.
