2,773 research outputs found
Partition genericity and pigeonhole basis theorems
There exist two notions of typicality in computability theory, namely,
genericity and randomness. In this article, we introduce a new notion of
genericity, called partition genericity, which is at the intersection of these
two notions of typicality, and show that many basis theorems apply to partition
genericity. More precisely, we prove that every co-hyperimmune set and every
Kurtz random is partition generic, and that every partition generic set admits
weak infinite subsets. In particular, we answer a question of Kjos-Hanssen and
Liu by showing that every Kurtz random admits an infinite subset which does not
compute any set of positive Hausdorff dimension. Partition genericty is a
partition regular notion, so these results imply many existing pigeonhole basis
theorems.Comment: 23 page
Open questions about Ramsey-type statements in reverse mathematics
Ramsey's theorem states that for any coloring of the n-element subsets of N
with finitely many colors, there is an infinite set H such that all n-element
subsets of H have the same color. The strength of consequences of Ramsey's
theorem has been extensively studied in reverse mathematics and under various
reducibilities, namely, computable reducibility and uniform reducibility. Our
understanding of the combinatorics of Ramsey's theorem and its consequences has
been greatly improved over the past decades. In this paper, we state some
questions which naturally arose during this study. The inability to answer
those questions reveals some gaps in our understanding of the combinatorics of
Ramsey's theorem.Comment: 15 page
Effective Genericity and Differentiability
We prove that a real x is 1-generic if and only if every differentiable
computable function has continuous derivative at x. This provides a counterpart
to recent results connecting effective notions of randomness with
differentiability. We also consider multiply differentiable computable
functions and polynomial time computable functions.Comment: Revision: added sections 6-8; minor correction
Genericity of nondegenerate geodesics with general boundary conditions
Let M be a possibly noncompact manifold. We prove, generically in the
C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on
M do not possess any degenerate geodesics satisfying suitable boundary
conditions. This extends a result of Biliotti, Javaloyes and Piccione for
geodesics with fixed endpoints to the case where endpoints lie on a compact
submanifold P of the product MxM that satisfies an admissibility condition.
Such condition holds, for example, when P is transversal to the diagonal of
MxM. Further aspects of these boundary conditions are discussed and general
conditions under which metrics without degenerate geodesics are C^k-generic are
given.Comment: LaTeX2e, 21 pages, no figure
Norms on possibilities I: forcing with trees and creatures
We present a systematic study of the method of "norms on possibilities" of
building forcing notions with keeping their properties under full control. This
technique allows us to answer several open problems, but on our way to get the
solutions we develop various ideas interesting per se.These include a new
iterable condition for ``not adding Cohen reals'' (which has a flavour of
preserving special properties of p-points), new intriguing properties of
ultrafilters (weaker than being Ramsey but stronger than p-point) and some new
applications of variants of the PP--property.Comment: accepted for Memoirs of the Amer. Math. So
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