Infinite Combinatorics: From Finite to Infinite

The infinite combinatorics here give statements in which, from some sequence, an infinite subsequence will satisfy some condition - for example, belong to some specified set. Our results give such statements generically - that is, for 'nearly all' points, or as we shall say, for quasi all points - all off a null set in the measure case, or all off a meagre set in the category case. The prototypical result here goes back to Kestelman in 1947 and to Borwein and Ditor in the measure case, and can be extended to the category case also. Our main result is what we call the Category Embedding Theorem, which contains the Kestelman-Borwein-Ditor Theorem as a special case. Our main contribution is to obtain function wise rather than point wise versions of such results. We thus subsume results in a number of recent and related areas, concerning e.g., additive, subadditive, convex and regularly varying functions.

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INFINITE COMBINATORICS PLAIN AND SIMPLE

Journal of Symbolic Logic ◽

10.1017/jsl.2018.8 ◽

2018 ◽

Vol 83 (3) ◽

pp. 1247-1281 ◽

Cited By ~ 2

Author(s):

DÁNIEL T. SOUKUP ◽

LAJOS SOUKUP

Keyword(s):

Chromatic Number ◽

Set Theory ◽

Wide Applicability ◽

Elementary Submodels ◽

Set Systems ◽

Point Set ◽

Infinite Combinatorics ◽

Sparse Set ◽

Paradoxical Decompositions ◽

General Method

AbstractWe explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

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Infinite Combinatorics and Large Cardinals

Perspectives in Mathematical Logic - Basic Set Theory ◽

10.1007/978-3-662-02308-2_9 ◽

1979 ◽

pp. 289-356

Author(s):

Azriel Levy

Keyword(s):

Large Cardinals ◽

Infinite Combinatorics

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