Archive for Mathematical Logic

, Volume 57, Issue 3–4, pp 381–389 | Cite as

A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem

  • Lorenzo CarlucciEmail author


Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \(\mathbf {RCA}_0\) to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman’s Theorem with apartness to the Increasing Polarized Ramsey’s Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman’s Theorem homogeneity is required only for finite sums of adjacent elements.


Hindman’s Theorem Computable combinatorics Ramsey’s Theorem Reverse mathematics 

Mathematics Subject Classification

05D10 03B30 03F35 


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  1. 1.
    Blass, A.: Some questions arising from Hindman’s Theorem. Sci. Math. Jpn. 62, 331–334 (2005)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Blass, A.R., Hirst, J.L., Simpson, S.G.: Logical analysis of some theorems of combinatorics and topological dynamics. In: Logic and Combinatorics (Arcata, Calif., 1985), Contemp. Math., vol. 65, pp. 125–156. American Mathematical Society, Providence, RI (1987)Google Scholar
  3. 3.
    Carlucci, L.: “Weak yet strong” restrictions of Hindman’s Finite Sums Theorem. Proc. Am. Math. Soc. (to appear)Google Scholar
  4. 4.
    Carlucci, L., Kołodziejczyk, L. A., Lepore, F., Zdanowski, K.: New bounds on the strength of some restrictions of Hindman’s Theorem. In: Kari, J., Manea, F., Petre, I. (eds.) Unveiling Dynamics and Complexity, 13th Conference Computability in Europe 2017 (Turku, Finland, June 12–16, 2017), Springer, pp. 210–220 (2017)Google Scholar
  5. 5.
    Cholak, P., Jockusch, C.G., Slaman, T.: On the strength of Ramsey’s Theorem for pairs. J. Symb. Log. 66(1), 1–55 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chong, C.T., Lempp, S., Yang, Y.: On the role of the collection principle for \(\Sigma ^0_2\) formulas in second-order reverse mathematics. Proc. Am. Math. Soc. 138, 1093–1100 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chong, C.T., Slaman, T.A., Yang, Y.: The metamathematics of the Stable Ramsey’s Theorem for Pairs. J. Am. Math. Soc. 27, 863–892 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dzhafarov, D.D., Hirst, J.L.: The polarized Ramsey’s theorem. Arch. Math. Log. 48(2), 141–157 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dzhafarov, D., Jockusch, C., Solomon, R., Westrick, L.B.: Effectiveness of Hindman’s Theorem for bounded sums. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A. (eds.) Proceedings of the International Symposium on Computability and Complexity (in Honour of Rod Downey’s 60th birthday). Lecture Notes in Computer Science, pp. 134–142. Springer (2017)Google Scholar
  10. 10.
    Hindman, N.: The existence of certain ultrafilters on \(N\) and a conjecture of Graham and Rothschild. Proc. Am. Math. Soc. 36(2), 341–346 (1972)zbMATHGoogle Scholar
  11. 11.
    Hindman, N.: Finite sums from sequences within cells of a partition of N. J. Comb. Theory Ser. A 17, 1–11 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hindman, N., Leader, I., Strauss, D.: Open problems in partition regularity. Comb. Probab. Comput. 12, 571–583 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hirst, J.: Hilbert vs Hindman. Arch. Math. Log. 51(1–2), 123–125 (2012)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kołodziejczyk, L.A., Michalewski, H., Pradic, P., Skrzypczak, M.: The logical strength of Büchi’s decidability theorem. In: Reigner, L., Talbot, J.-M. (eds.) Proceedings of Computer Science Logic 2016, CSL 2016, August 29 to September 1, 2016, Marseille, France. Leibniz International Proceedings in Informatics, pp. 36:1–36:16. Dagstuhl Publishing (2016)Google Scholar
  15. 15.
    Montalbán, A.: Open questions in reverse mathematics. Bull. Symb. Log. 17(3), 431–454 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Patey, L., Yokoyama, K.: The proof theoretic strength of Ramsey’s Theorem for pairs. Preprint (2016)Google Scholar
  17. 17.
    Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Logic, 2nd edn. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Dipartimento di InformaticaSapienza — Università di RomaRomeItaly

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