Notre Dame Journal of Formal Logic

A Simple Proof and Some Difficult Examples for Hindman's Theorem

Henry Towsner


We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several examples of colorings of the integers which do not have computable witnesses to Hindman's Theorem.

Article information

Notre Dame J. Formal Logic, Volume 53, Number 1 (2012), 53-65.

First available in Project Euclid: 9 May 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]
Secondary: 05D10: Ramsey theory [See also 05C55]

Hindman's Theorem reverse mathematics Ramsey theory


Towsner, Henry. A Simple Proof and Some Difficult Examples for Hindman's Theorem. Notre Dame J. Formal Logic 53 (2012), no. 1, 53--65. doi:10.1215/00294527-1626518.

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  • Baumgartner, J. E., "A short proof of Hindman's theorem", Journal of Combinatorial Theory. Series A, vol. 17 (1974), pp. 384–86.
  • Blass, A. R., J. L. Hirst, and S. G. Simpson, "Logical analysis of some theorems of combinatorics and topological dynamics", pp. 125–56 in Logic and Combinatorics (Arcata, CA, 1985), vol. 65 of Contemporary Mathematics, American Mathematical Society, Providence, 1987.
  • Comfort, W. W., "Ultrafilters: Some old and some new results", Bulletin of the American Mathematical Society, vol. 83 (1977), pp. 417–55.
  • Hindman, N., "Finite sums from sequences within cells of a partition of $N$", Journal of Combinatorial Theory. Series A, vol. 17 (1974), pp. 1–11.
  • Hindman, N., "Algebra in the Stone-Čech compactification and its applications to Ramsey theory", Scientiae Mathematicae Japonicae, vol. 62 (2005), pp. 321–29.
  • Hindman, N., and D. Strauss, Algebra in the Stone-Čech Compactification. Theory and Applications, vol. 27 of de Gruyter Expositions in Mathematics, Walter de Gruyter & Co., Berlin, 1998.
  • Simpson, S. G., Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1999.
  • Towsner, H., "Hindman's Theorem: An ultrafilter argument in second order arithmetic", The Journal of Symbolic Logic, vol. 76 (2011), pp. 353–60.