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Infinite combinatorics. (English) Zbl 0844.03026

Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 1-2. Amsterdam: Elsevier (North-Holland). 2085-2116 (1995).
This is a vivid, nicely written exposition of the most important tools and results of infinitary combinatorics. Carefully selected topics introduce us to the art of treating problems which arise from generalizing finite questions to the infinite.
The detailed contents are as follows.
First those graph invariants are studied which are defined as maxima in the finite case (like the maximum degree). Then several formulations of the König lemma are given and a description of the compactness principle is shown. This is followed by a part on Ramsey type results which contains the most important positive and negative results. After this, trees, large cardinals (strongly inaccessible, weakly compact, measurable) are introduced with their connections to partition problems.
Finally, some classical topics on uncountably chromatic graphs are covered like obligatory subgraphs and examples with the chromatic number jumping. The chapter ends with a short introduction to axiomatic set theory quoting results on cofinality, ultrafilters, closed unbounded, and stationary sets.
For the entire collection see [Zbl 0833.05001].

MSC:

03E05 Other combinatorial set theory
03E55 Large cardinals
05C55 Generalized Ramsey theory
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
05C15 Coloring of graphs and hypergraphs
05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
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