Coloring Problems for Infinite Words

  • Caïus Wojcik
  • Luca Q. Zamboni
Part of the Trends in Mathematics book series (TM)


Given a finite coloring (or finite partition) of the free semigroup \({\mathcal A}^+\) over a set \({\mathcal A}\), we consider various types of monochromatic factorizations of right-sided infinite words \(x\in {\mathcal A}^\omega \). In 2006 T. Brown asked the following question in the spirit of Ramsey theory: Given a nonperiodic infinite word x = x1x2x3⋯ with values in a set \({\mathcal A} \), does there exist a finite Coloring \(\varphi : {\mathcal A} ^+\rightarrow C\) relative to which x does not admit a φ-monochromatic factorization, i.e., a factorization of the form x = u1u2u3⋯ with φ(u i ) = φ(u j ) for all i, j ≥ 1? We give an optimal affirmative answer to this question by showing that if x = x1x2x3⋯ is an infinite nonperiodic word with values in a set \({\mathcal A},\) then there exists a 2-coloring \(\varphi : {\mathcal A} ^+\rightarrow \{0,1\}\) such that for any factorization x = u1u2u3⋯, we have φ(u i ) ≠ φ(u j ) for some i ≠ j. Some stronger versions of the usual notion of monochromatic factorization are also introduced and studied. We establish links, and in some cases equivalences, between the existence of these factorizations and fundamental results in Ramsey theory including the infinite Ramsey theorem, Hindman’s finite sums theorem, partition regularity of IP-sets, and the Milliken–Taylor theorem.


  1. 14.
    Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)Google Scholar
  2. 47.
    Baumgartner, J.E.: A short proof of Hindman’s theorem. J. Comb. Theory Ser. A 17, 384–386 (1974)MathSciNetCrossRefGoogle Scholar
  3. 73.
    Bernardino, A., Pacheco, R., Silva, M.: Coloring factors of substitutive infinite words (2016). ArXiv:1605.09343Google Scholar
  4. 110.
    Brown, T.C.: Colorings of the factors of a word (2006). Preprint, Department of Mathematics, Simon Fraser University, CanadaGoogle Scholar
  5. 205.
    Durand, F., Host, B., Skau, C.: Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergodic Theory Dyn. Syst. 19(4), 953–993 (1999)MathSciNetCrossRefGoogle Scholar
  6. 290.
    Hindman, N.: Finite sums of sequences within cells of a partition of \({\mathbb N}\). J. Comb. Theory. Ser. A 17, 1–11 (1974)Google Scholar
  7. 291.
    Hindman, N.: Partitions and sums of integers with repetition. J. Comb. Theory. Ser. A 27, 19–32 (1979)MathSciNetCrossRefGoogle Scholar
  8. 292.
    Hindman, N., Leader, I., Strauss, D.: Pairwise sums in colourings of the reals. Abh. Math. Semin. Univ. Hambg. 1–13 (2016)Google Scholar
  9. 293.
    Hindman, N., Strauss, D.: Algebra in the Stone-Čech compactification: theory and applications. Walter de Gruyter, Berlin (2012)Google Scholar
  10. 385.
    Lothaire, M.: Combinatorics on Words. Encyclopedia of Mathematics and Its Applications, vol. 17. Addison-Wesley, Reading (1983)Google Scholar
  11. 390.
    de Luca, A., Pribavkina, E., Zamboni, L.: A coloring problem for infinite words. J. Comb. Theory. Ser. A 125, 306–332 (2014)Google Scholar
  12. 392.
    de Luca, A., Zamboni, L.: On prefixal factorisations of words. Eur. J. Comb. 52, 59–73 (2016)Google Scholar
  13. 393.
    de Luca, A., Zamboni, L.: On some variations of coloring problems of infinite words. J. Comb. Theory. Ser. A 137, 166–178 (2016)Google Scholar
  14. 417.
    Milliken, K.: Ramsey’s theorem with sums or unions. J. Comb. Theory Ser. A 18, 276–290 (1975)MathSciNetCrossRefGoogle Scholar
  15. 448.
    Niven, I.: Irrational Numbers. MAA (1963)Google Scholar
  16. 464.
    Owings, J.: Problem e2494. Am. Math. Mon. 81 (1974)Google Scholar
  17. 491.
    Ramsey, F.P.: On a problem of formal logic. Proc. Lond. Math. Soc. 30, 264–286 (1930)MathSciNetCrossRefGoogle Scholar
  18. 519.
    Salo, V., Törmä, I.: Factor colorings of linearly recurrent words (2015). ArXiv:1504.0582Google Scholar
  19. 532.
    Schützenberger, M.P.: Quelques problèmes combinatoires de la théorie des automates. Cours professé à l’Institut de Programmation (1999). J.-F. PerrotGoogle Scholar
  20. 548.
    Sierpiński, W.: Sur un problème de la théorie des relations. Ann. Scuola Norm. Sup. Pisa 2, 285–287 (1933)MathSciNetzbMATHGoogle Scholar
  21. 561.
    Taylor, A.: A canonical partition relation for finite subsets of ω. J. Comb. Theory. Ser. A 21, 137–146 (1976)Google Scholar
  22. 562.
    Thue, A.: Über unendliche Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 7, 1–22 (1906). Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor, Universitetsforlaget, Oslo, 1977, pp. 139–158Google Scholar
  23. 563.
    Thue, A.: Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1, 1–67 (1912). Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor, Universitetsforlaget, Oslo, 1977, pp. 413–478Google Scholar
  24. 589.
    Wojcik, C., Zamboni, L.Q.: Monochromatic factorisations of words and periodicity (2017). PreprintGoogle Scholar
  25. 591.
    Zamboni, L.Q.: A note on coloring factors of words (2010). In: Oberwolfach Report 37/2010, Mini-workshop: Combinatorics on Words, August, 22–27Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut Camille JordanUniversité Lyon 1, CNRS UMR 5208Villeurbanne CedexFrance

Personalised recommendations