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.
Research supported by Hungarian National OTKA Fund No. 2117.
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Bibliography
U. Abraham: Free sets for nowhere-dense set mappings, Israel Journal of Mathematics, 39 (1981), 167–176.
F. Bagemihl: The existence of an everywhere dense independent set, Michigan Mathematical Journal 20 (1973), 1–2.
J. E.Baumgartner: Partitioning vector spaces, J. Comb. Theory, Ser. A. 18 (1975),231–233.
J. Ceder: Finite subsets and countable decompositions of Euclidean spaces, Rev. Roumaine Math. Pures Appl. 14 (1969), 1247–1251.
Z. Daróczy: Jelentés az 1965. évi Schweitzer Miklós matematikai emlékversenyről, Matematikai Lapok 17 (1966), 344–366. (in Hungarian)
R. O. Davies: Covering the plane with denumerably many curves, Bull. London Math. Soc. 38 (1963), 343–348.
R. O. Davies: Partitioning the plane into denumerably many sets without repeated distances, Proc. Camb. Phil. Soc. 72 (1972), 179–183.
R. O. Davies: Covering the space with denumerably many curves, Bull. London Math. Soc. 6 (1974), 189–190.
P. Erdős: Some remarks on set theory, Proc. Amer. Math. Soc. 1 (1950), 127–141.
P. Erdős: Some remarks on set theory, III, Michigan Mathematical Journal 2 (1953–54), 51–57.
P. Erdős: Some remarks on set theory IV, Mich. Math. 2 (1953–54), 169–173.
P. Erdős: Hilbert terben levő ponthalmazok néhány geometriai és halmazelméleti tulajdonságáról, Matematikai Lapok 19 (1968), 255–258 (Geometrical and set theoretical properties of subsets of Hilbert spaces, in Hungarian).
P. Erdős: Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange 4 (1978–79), 113–138.
P. Erdős: My Scottish Book “Problems”, in: The Scottish Book, Mathematics from the Scottish Café (ed. R.D.Mauldin), Birkhauser, 1981,35–43.
P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, E. G. Straus: Euclidean Ramsey theorems II, in: Infinite and Finite Sets, Keszthely (Hungary), 1973, ColI. Math. Soc. J. Bolyai 10, 529–557.
P. Erdős, A. Hajnal: Some remarks on set theory, VIII, Michigan Mathematical Journal 7 (1960), 187–191.
P. Erdős, S. Jackson, R. D. Mauldin: On partitions of lines and space, Fund. Math. 145 (1994), 101–119.
P. Erdős, S. Kakutani: On non-denumerable graphs, Bull. Amer. Math. Soc., 49 (1943), 457–461.
P. Erdős, P. Komjáth: Countable decompositions of R 2 and R 3, Discrete and Computational Geometry 5 (1990), 325–331.
C. Freiling: Axioms of symmetry: Throwing darts at the real number line, Journal of Symbolic Logic 51 (1986), 190–200.
H. Friedman: A consistent Fubini-Tonelli theorem for nonmeasurable functions, Illinois Journal of Mathematics, 24 (1980), 390–395.
S. H. Hechler: Directed graphs over topological spaces: some set theoretical aspects, Israel Journal of Mathematics 11 (1972), 231–248.
P. Komjáth: Tetrahedron free decomposition of R 3, Bull. London Math. Soc. 23 (1991), 116–120.
P. Komjáth: The master coloring, Comptes Rendus Mathématiques de l’Academie des sciences, la Société royale du Canada, 14(1992), 181–182.
P. Komjáth: Set theoretic constructions in Euclidean spaces, in : New trends in Discrete and Computational Geometry, (J. Pach, ed.), Springer, Algorithms and Combinatorics, 10 (1993), 303–325.
P. Komjáth: A decomposition theorem for R n, Proc. Amer. Math. Soc. 120 (1994), 921–927.
P. Komjáth: Partitions of vector spaces, Periodica Math. Hung. 28 (1994), 187–193.
P. Komjáth: A note on set mappings with meager images, Studia Math. Hung., accepted.
K. Kunen: Partitioning Euclidean space, Math. Proc. Camb. Phil. Soc. 102, (1987), 379–383.
L. Newelski, J. Pawlikowski, W. Seredyńiski: Infinite free set for small measure set mappings, Proc. Amer. Math. Soc. 100 (1987), 335–339.
J. H. Schmerl: Partitioning Euclidean space, Discrete and Computational Geometry
J. H. Schmerl: Triangle-free partitions of Euclidean space, to appear.
J. H. Schmerl: Countable partitions of Euclidean space, to appear.
W. Sierpińiski: Sur un théorèms équivalent à l’hypothese du continu, Bull. Int. Acad. Sci. Cracovie A (1919), 1–3.
W. Sierpińiski: Hypothèse du continu, Warsaw, 1934.
J. C. Simms: Sierpińiski’s theorem, Simon Stevin 65 (1991), 69–163.
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Komjáth, P. (2013). Set Theory: Geometric and Real. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős II. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7254-4_24
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