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Infinite Combinatorics: From Finite to Infinite

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Horizons of Combinatorics

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 17))

Abstract

We investigate the relationship between some theorems in finite combinatorics and their infinite counterparts: given a “finite” result how one can get an “infinite” version of it? We will also analyze the relationship between the proofs of a “finite” theorem and the proof of its “infinite” version.

The preparation of this paper was supported by the Hungarian National Foundation for Scientific Research grant no. 61600 and 68262

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References

  1. R. Aharoni and E. Berger, Menger’s Theorem for Infinite Graphs, J. Graph Theory, 50 (2005), 199–211.

    Article  MathSciNet  Google Scholar 

  2. R. Aharoni, E. C. Milner and K. Prikry, Unfriendly partitions of a graph, J. Corn-bin. Theory, Ser. B, 50 (1990), no. 1, 1–10.

    Article  MATH  MathSciNet  Google Scholar 

  3. R. Ahlswede, P. L. Erdős and N. Graham, A splitting property of maximal antichains, Combinatorica, 15 (1995), no. 4, 475–480.

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Bollobás, Modern graph theory, Graduate Texts in Mathematics, 184, Springer-Verlag (New York, 1998).

    MATH  Google Scholar 

  5. V. Chvátal and L. Lovász, Every directed graph has a semi-kernel, Hypergraph Seminar (Proc. First Working Sem., Ohio State Univ., Columbus, Ohio, 1972; dedicated to Arnold Ross), pp. 175. Lecture Notes in Math., Vol. 411, Springer (Berlin, 1974).

    Chapter  Google Scholar 

  6. E. Dalhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour and M. Yannakakis, The Complexity of Multiway Cuts, Proc. 24th Annual ACM Symp. on Theory of Computing (1992), pp. 241–251.

    Google Scholar 

  7. R. Diestel, Graph theory, Third edition. Graduate Texts in Mathematics, 173. Springer-Verlag (Berlin, 2005).

    MATH  Google Scholar 

  8. R. Diestel, Directions in Infinite Graph Theory and Combinatorics, Topics in Discrete Mathematics 3, Elsevier — North Holland, 1992.

    Google Scholar 

  9. P. Erdős, T. Grünwald and E. Vázsonyi, Végtelen gráfok Euler vonalairól (On Euler lines of infinite graphs, in Hungarian), Mat. Fiz. Lapok, 43 (1936).

    Google Scholar 

  10. P. Erdős, T. Grünwald and E. Vázsonyi, Über Euler-Linien unendlicher Graphen, J. Math. Phys., Mass. Inst. Techn., 17 (1938), 59–75.

    Google Scholar 

  11. P. L. Erdős, A. Frank and L. A. Székely, Minimum multiway cuts in trees, Discrete Appl. Math., 87 (1998), no. 1–3, 67–75.

    Article  MathSciNet  Google Scholar 

  12. P. L. Erdős and L. Soukup, How to split antichains in infinite posets, Combinatorica, 27 (2007), no. 2, 147–161.

    Article  MathSciNet  Google Scholar 

  13. P. L. Erdős and L. Soukup, Quasi-kernels and quasi-sinks in infinite graphs, submitted.

    Google Scholar 

  14. P. L. Erdős and L. A. Székely, Evolutionary trees: an integer multicommodity max-flow-min-cut theorem, Adv. in Appl. Math., 13 (1992), no. 4, 375–389.

    Article  MathSciNet  Google Scholar 

  15. A. Hajnal, Infinite combinatorics, Handbook of combinatorics, Vol. 1,2, 2085–2116, Elsevier (Amsterdam, 1995).

    Google Scholar 

  16. A. Hajnal, The chromatic number of the product of two N1-chromatic graphs can be countable, Combinatorica, 5 (1985), no. 2, 137–139.

    Article  MATH  MathSciNet  Google Scholar 

  17. P. Komjáth and V. Totik, Problems and theorems in classical set theory, Problem Books in Mathematics, Springer (New York, 2006).

    MATH  Google Scholar 

  18. S. Shelah and E. C. Milner, Graphs with no unfriendly partitions, A tribute to Paul Erdős, Cambridge Univ. Press (Cambridge, 1990), pp. 373–384.

    Google Scholar 

  19. L. Soukup, On chromatic number of product of graphs, Comment. Math. Univ. Carolin., 29 (1988), no. 1, 1–12.

    MATH  MathSciNet  Google Scholar 

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© 2008 János Bolyai Mathematical Society and Springer-Verlag

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Soukup, L. (2008). Infinite Combinatorics: From Finite to Infinite. In: Győri, E., Katona, G.O.H., Lovász, L., Sági, G. (eds) Horizons of Combinatorics. Bolyai Society Mathematical Studies, vol 17. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77200-2_10

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