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INFINITE COMBINATORICS PLAIN AND SIMPLE

Published online by Cambridge University Press:  23 October 2018

DÁNIEL T. SOUKUP and
LAJOS SOUKUP
DÁNIEL T. SOUKUP
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITÄT WIEN, WIEN, AUSTRIAE-mail: daniel.soukup@univie.ac.atURL: http://www.logic.univie.ac.at/∼soukupd73/
LAJOS SOUKUP
Affiliation:
HUNGARIAN ACADEMY OF SCIENCES ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS, BUDAPEST, HUNGARYE-mail: soukup@renyi.huURL: http://www.renyi.hu/∼soukup
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Abstract

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We 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.

Keywords

03E0503C9805C6303E3554A35elementary submodelsDavies-treecloudschromatic numberalmost disjointBernsteinCantorcoloringsplendidcountably closed

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 3 , September 2018, pp. 1247 - 1281
Copyright
Copyright © The Association for Symbolic Logic 2018 

References

REFERENCES

Arhangel’skii, A. V., On the cardinality of bicompacta satisfying the first axiom of countability. Doklady Akademii Nauk, vol. 10 (1969), pp. 951–955.Google Scholar
Bernstein, F., Zur Theorie der trigonometrischen Reihen, Sitzungsber. Sächs. Akad. Wiss. Leipzig. Math.-Natur. Kl., vol. 60 (1908), pp. 325–338.Google Scholar
Cheilaris, P., Conflict-free coloring, Ph.D. thesis, City University of New York, 2008.Google Scholar
Davies, R. O., Covering the plane with denumerably many curves. Journal of the London Mathematical Society, vol. 38 (1963), pp. 433–438.CrossRefGoogle Scholar
Dow, A., An introduction to applications of elementary submodels to topology. Topology Proceedings, vol. 13 (1988), no. 1, pp. 17–72.Google Scholar
Dow, A., Set Theory in Topology, Recent Progress in General Topology (Prague, 1991), North-Holland, Amsterdam, 1992, pp. 167–197.Google Scholar
Van Douwen, E., The integers and topology, Handbook of Set-Theoretic Topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984, pp. 111–167.CrossRefGoogle Scholar
Erdős, P. and Hajnal, A., On chromatic number of graphs and set-systems. Acta Mathematica Hungarica, vol. 17 (1966), pp. 61–99.CrossRefGoogle Scholar
Erdős, P. and Hajnal, A., Unsolved and solved problems in set theory, Proceedings of the Tarski Symposium (Henkin, L., Addison, J., Chang, C. C., Craig, W., Scott, D., and Vaught, R., editors), Proceedings of Symposia in Pure Mathematics, vol. XXV, American Mathematical Society, Providence, RI, 1974, pp. 269–287.CrossRefGoogle Scholar
Erdős, P. and Shelah, S., Separability properties of almost—Disjoint families of sets. Israel Journal of Mathematics, vol. 12, no. 2, (1972), pp. 207–214.CrossRefGoogle Scholar
Eskew, M. and Hayut, Y., On the consistency of local and global versions of Chang’s Conjecture. Transactions of the American Mathematical Society, vol. 370 (2018), no. 4, pp. 2879–2905.CrossRefGoogle Scholar
Even, G., Lotker, Z., Ron, D., and Smorodinsky, S., Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellural networks. Society of Indian Automobile Manufactures Journal on Computing, vol. 33 (2003), pp. 94–136.Google Scholar
Foreman, M. and Magidor, M., A very weak square principle, this JOURNAL, vol. 62 (1997), no. 1, pp. 175–196.Google Scholar
Fuchino, S. and Soukup, L., More set-theory around the weak Freese-Nation property, European Summer Meeting of the Association for Symbolic Logic (Haifa, 1995). Fundamenta Mathematicae, vol. 154 (1995), no. 2, pp. 159–176.Google Scholar
Geschke, S., Applications of elementary submodels in general topology, Foundations of the formal sciences, 1 (Berlin, 1999). Synthese, vol. 133 (2002), no. 1–2, pp. 31–41.CrossRefGoogle Scholar
Goldstern, M., Haim, J., and Shelah, S., Saturated families. Proceedings of the American Mathematical Society, vol. 111 (1991), no. 4, pp. 1095–1104.CrossRefGoogle Scholar
Hajnal, A., Juhász, I., and Shelah, S., Splitting strongly almost disjoint families. Transactions of the American Mathematical Society, vol. 295 (1986), no. 1, pp. 369–387.CrossRefGoogle Scholar
Hajnal, A., Juhász, I., Soukup, L., and Szentmiklóssy, Z., Conflict free colorings of (strongly) almost disjoint set-systems. Acta Mathematica Hungarica, vol. 131 (2010), no. 3, pp. 230–274.CrossRefGoogle Scholar
Jackson, S. and Mauldin, R., On a lattice problem of H. Steinhaus. Journal of the American Mathematical Society, vol. 15 (2002), no. 4, pp. 817–856.CrossRefGoogle Scholar
Jackson, S. and Mauldin, R., Survey of the Steinhaus Tiling Problem. The Bulletin of Symbolic Logic, vol. 9 (2003), no. 3, pp. 335–361.CrossRefGoogle Scholar
Jech, T., Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
Juhász, I., Nagy, Zs., and Weiss, W., On countably compact, locally countable spaces. Periodica Mathematica Hungarica, vol. 10 (1979), no. (2–3), pp. 193–206.CrossRefGoogle Scholar
Juhasz, I., Shelah, S., and Soukup, L., More on countably compact, locally countable spaces. Israel Journal of Mathematics, vol. 62 (1988), no. 3, pp. 302–310.CrossRefGoogle Scholar
Juhász, I. and Weiss, W., Good, splendid and Jakovlev, Open Problems in Topology II (Pearl, E., editor), Elsevier, Amsterdam, 2007, pp. 185–188.CrossRefGoogle Scholar
Just, W. and Weese, M., Discovering modern set theory. II, Set-theoretic Tools for Every Mathematician, Graduate Studies in Mathematics, vol. 18, American Mathematical Society, Providence, RI, 1997.Google Scholar
Kojman, M., Splitting families of sets in ZFC. Advances in Mathematics, vol. 269 (2015), pp. 707–725.CrossRefGoogle Scholar
Komjáth, P., Families close to disjoint ones. Acta Mathematica Hungarica, vol. 43 (1984), no. 3–4, pp. 199–207.CrossRefGoogle Scholar
Komjáth, P., Connectivity and chromatic number of infinite graphs. Israel Journal of Mathematics, vol. 56 (1986), no. 3, 257–266.CrossRefGoogle Scholar
Komjáth, P., The colouring number. Proceedings of the London Mathematical Society (3), vol. 54 (1987), no. 1, pp. 1–14.CrossRefGoogle Scholar
Komjáth, P., Three clouds may cover the plane. Annals of Pure and Applied Logic, vol. 109 (2001), pp. 71–75.CrossRefGoogle Scholar
Komjáth, P., The chromatic number of infinite graphs—A survey. Discrete Mathematics, vol. (2011), no. 15, pp. 1448–1450.CrossRefGoogle Scholar
Kunen, K., Set Theory, Studies in Logic, vol. 34, College Publications, London.Google Scholar
Levinski, J.-P., Magidor, M., and Shelah, S.. Chang’s conjecture for ${\aleph _\omega }$.. Israel Journal of Mathematics, vol. 69 (1990), no. 2, pp. 161–172.CrossRefGoogle Scholar
Milovich, D., Noetherian types of homogeneous compacta and dyadic compacta. Topology and its Applications, vol. 156 (2008), pp. 443–464.CrossRefGoogle Scholar
Milovich, D., The $\left( {\lambda ,\kappa } \right)$-Freese-Nation property for Boolean algebras and compacta. Order, vol. 29 (2012), no. 2, pp. 361–379.CrossRefGoogle Scholar
Milovich, D., On the Strong Freese-Nation property. Order, vol. 34 (2017), no. 1, pp. 91–111.CrossRefGoogle Scholar
Milovich, D., Amalgamating many overlapping Boolean algebras, preprint, 2017, arXiv:1607.07944.Google Scholar
Mycielski, J., Sur le coloriage des graphs. Colloquium Mathematicum, vol. 3, (1955), pp. 161–162.CrossRefGoogle Scholar
Pach, J. and Tóth, G., Conflict-free colorings, Discrete and Computational Geometry: The Goodman-Pollack Festschrift (Aronov, B., Basu, S., Pach, J., and Sharir, M., editors), Algorithms and Combinatorics, vol. 25, Springer-Verlag, Berlin, 2003, pp. 665–671.CrossRefGoogle Scholar
Pach, J. and Tardos, G., Conflict-free colorings of graphs and hypergraphs. Combinatorics, Probability and Computing, vol. 18 (2009), pp. 819–834.CrossRefGoogle Scholar
Schmerl, J. H., How many clouds cover the plane? Fundamenta Mathematicae, vol. 177 (2003), no. 3, pp. 209–211.CrossRefGoogle Scholar
Shelah, S., Anti-homogeneous partitions of a topological space. Scientiae Mathematicae Japonicae, vol. 59 (2004), no. 2, pp. 203–256.Google Scholar
Shelah, S., MAD saturated families and SANE player. Canadian Journal of Mathematics, vol. 63 (2011), no. 6, pp. 1416–1436.CrossRefGoogle Scholar
Sierpinski, W., Sur le recouvrement du plan par une infinite denombrable de courbes congiuentes. Fundamenta Mathematicae, vol. 21 (1933), pp. 39–42.CrossRefGoogle Scholar
Sierpinski, W., Hypothèse du Continu, Monografje Matematyczne, Tome 4, Warszawa-Lwow, Subwencji Funduszu Kultur, 1934.Google Scholar
Soukup, D. T., Trees, ladders and graphs. Journal of Combinatorial Theory, Series B, vol. 115 (2015), pp. 96–116.CrossRefGoogle Scholar
Soukup, L., Elementary submodels in infinite combinatorics. Discrete Mathematics, vol. 311 (2011), no. 15, pp. 1585–1598.CrossRefGoogle Scholar
Todorcevic, S., Walks on Ordinals and their Characteristics, Progress in Mathematics, vol. 263, Birkhäuser Verlag, Basel, 2007.Google Scholar
Velleman, D., Simplified morasses, this JOURNAL, vol. 49 (1984), no. 1, pp. 257–271.Google Scholar
Weiss, W., Partitioning topological spaces, Topology, Vol. II (Proceedings of the Fourth colloquial, Budapest, 1978) (Császár, Á., editor), Colloquia mathematica Societatis János Bolyai, vol. 23, North-Holland, Amsterdam, 1980.Google Scholar
Weiss, W., Partitioning topological spaces, Mathematics of Ramsey Theory (Nešetřil, J. and Rödl, V., editors), Algorithms and Combinatorics, vol. 5, Springer-Verlag, Berlin, 1990, pp. 154–171.CrossRefGoogle Scholar