# Set Theory: Geometric and Real

• Péter Komjáth
Chapter
Part of the Algorithms and Combinatorics book series (AC, volume 14)

## Abstract

In this Chapter we consider P. Erdős’ research on what can be called as the borderlines of set theory with some of the more classical branches of mathematics as geometry and real analysis. His continuing interest in these topics arose from the world view in which the prime examples of sets are those which are subsets of some Euclidean spaces. ‘Abstract’ sets of arbitrary cardinality are of course equally existing. Paul only uses his favorite game for inventing new problems; having solved some problems find new ones by adding and/or deleting some structure on the sets currently under research. A good example is the one about set mappings. This topic was initiated by P. Turán who asked if a finite set f(x) is associated to every point x of the real line does necessarily exist an infinite free set, i.e., when xf(y) holds for any two distinct elements. Clearly the underlying structure has nothing to do with the question and eventually a nice theory emerged which culminated in the results of Erdős, G. Fodor, and A. Hajnal. But Paul and his collaborators kept returning to the original setup when the condition is e.g. changed to: let f(x) be nowhere dense, etc. Several nice and hard results have recently been proved. (See Section 8. in this Chapter.)

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