17,739 research outputs found
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
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
The weakness of being cohesive, thin or free in reverse mathematics
Informally, a mathematical statement is robust if its strength is left
unchanged under variations of the statement. In this paper, we investigate the
lack of robustness of Ramsey's theorem and its consequence under the frameworks
of reverse mathematics and computable reducibility. To this end, we study the
degrees of unsolvability of cohesive sets for different uniformly computable
sequence of sets and identify different layers of unsolvability. This analysis
enables us to answer some questions of Wang about how typical sets help
computing cohesive sets.
We also study the impact of the number of colors in the computable
reducibility between coloring statements. In particular, we strengthen the
proof by Dzhafarov that cohesiveness does not strongly reduce to stable
Ramsey's theorem for pairs, revealing the combinatorial nature of this
non-reducibility and prove that whenever is greater than , stable
Ramsey's theorem for -tuples and colors is not computably reducible to
Ramsey's theorem for -tuples and colors. In this sense, Ramsey's
theorem is not robust with respect to his number of colors over computable
reducibility. Finally, we separate the thin set and free set theorem from
Ramsey's theorem for pairs and identify an infinite decreasing hierarchy of
thin set theorems in reverse mathematics. This shows that in reverse
mathematics, the strength of Ramsey's theorem is very sensitive to the number
of colors in the output set. In particular, it enables us to answer several
related questions asked by Cholak, Giusto, Hirst and Jockusch.Comment: 31 page
Random processes via the combinatorial dimension: introductory notes
This is an informal discussion on one of the basic problems in the theory of
empirical processes, addressed in our preprint "Combinatorics of random
processes and sections of convex bodies", which is available at ArXiV and from
our web pages.Comment: 4 page
Counting faces of randomly-projected polytopes when the projection radically lowers dimension
This paper develops asymptotic methods to count faces of random
high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have
surprising implications - in statistics, probability, information theory, and
signal processing - with potential impacts in practical subjects like medical
imaging and digital communications. Three such implications concern: convex
hulls of Gaussian point clouds, signal recovery from random projections, and
how many gross errors can be efficiently corrected from Gaussian error
correcting codes.Comment: 56 page
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