Arithmetic Progressions and Tic-Tac-Toe Games

  • József Beck
Part of the Developments in Mathematics book series (DEVM, volume 16)


This paper is partly an overview of the subject (see Sections 1–4), in fact, as far as I know, the first attempt to do that, and partly an ordinary research paper containing proofs for new results (Sections 5–8). I use many different sources; to make the reader’s life easier, I decided to keep the paper (more-or-less) self-contained - this explains the considerable length.


Arithmetic progression probabilistic method derandomization strategy 

2000 Mathematics subject classification



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  1. 1.
    Alon, N., Spencer, J.: The Probabilistic Method. Academic Press, New York (1992)zbMATHGoogle Scholar
  2. 2.
    Baumgartner, J., Galvin, F., Laver, R., McKenzie, R.: Game theoretic versions of partition relations. In: Hajnal, A., Rado, R., Sós, V. T. (eds.) Infinite and Finite Sets. Colloq. Math. Soc. János Bolyai, vol. 10, pp. 131–135. North-Holland, Amsterdam (1973)Google Scholar
  3. 3.
    Beck, J.: On positional games. J. Comb. Theory, Ser. A 30, 117–133 (1981)zbMATHCrossRefGoogle Scholar
  4. 4.
    Beck, J.: Van der Waerden and Ramsey type games. Combinatorica 2, 103–116 (1981)CrossRefGoogle Scholar
  5. 5.
    Beck, J.: An algorithmic approach to the Lovász Local Lemma. I. Random Struct. Algorithms 2, 343–365 (1991)zbMATHCrossRefGoogle Scholar
  6. 6.
    Beck, J.: Positional games and the second moment method. Combinatorica 22, 169–216 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Berlekamp, E.R.: A construction for partitions which avoid long arithmetic progressions. Can. Math. Bull. 11, 409–414 (1968)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Berlekamp, E.R., Conway, J.H., Guy, R.K.: Winning Ways, vol. I and II. Academic Press, London (1982)zbMATHGoogle Scholar
  9. 9.
    Erdős, P.: Some remarks on the theory of graphs. Bull. Am. Math. Soc. 53, 292–294 (1947)CrossRefGoogle Scholar
  10. 10.
    Erdős, P.: On a combinatorial problem, I. Nordisk Mat. Tidskr. 11, 5–10 (1963)MathSciNetGoogle Scholar
  11. 11.
    Erdős, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. In: Hajnal, A., Rado, R., Sós, V. T. (eds.) Infinite and Finite Sets. Colloq. Math. Soc. Janos Bolyai, vol. 10, pp. 609–627. North-Holland, Amsterdam (1975)Google Scholar
  12. 12.
    Erdős, P., Selfridge, J.: On a combinatorial game. J. Comb. Theory Sen A 14, 298–301 (1973)CrossRefGoogle Scholar
  13. 13.
    Golomb, S.W., Hales, A.W.: Hypercube tic-tac-toe. In: Nowakowski, R. J. (ed.) More on Games of No Chance. Math. Sci. Res. Inst. Publ., vol. 42, pp. 167–182. Cambridge University Press, Cambridge (2002)Google Scholar
  14. 14.
    Gowers, T.: A new proof of Szemerédi’s theorem. Geom. Funct. Anal, 11, 465–588 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Graham, R.L., Rothschild, B.L., Spencer, J.H.: Ramsey Theory. Wiley-Interscience Series in Discrete Mathematics. Wiley, New York (1980)zbMATHGoogle Scholar
  16. 16.
    Hales, A.W, Jewett, R.I.: On regularity and positional games. Trans. Am. Math. Soc. 106, 222–229 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Patashnik, O.: Qubic: 4 x 4 x 4 tic-tac-toe. Math. Mag. 53, 202–216 (1980)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Schmidt, W.M.: Two combinatorial theorems on arithmetic progressions. Duke Math. J. 29, 129–140 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Shelah, S.: Primitive recursive bounds for van der Waerden numbers. J. Am. Math. Soc. 1, 683–697 (1988)zbMATHCrossRefGoogle Scholar
  20. 20.
    van der Waerden, B.L.: Beweis einer Baudetschen Vermutung. Nieu Arch. Wiskd. 15, 212–216 (1927)Google Scholar
  21. 21.
    Zermelo, E.: Über eine Anwendung der Mengenlehre und der Theorie des Schachspiels. In: Proceedings of the Fifth International Congress of Mathematicians, vol. 2, pp. 501–504. Cambridge University Press, Cambridge (1913)Google Scholar

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© Springer-Verlag 2008

Authors and Affiliations

  • József Beck
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

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