Using Ultrafilters to Prove Ramsey-type Theorems

References

• Ax, J. (1968). The elementary theory of finite fields. Ann. of Math. (2). 88(2): 239–271. DOI: https://doi.org/10.2307/1970573.
   Web of Science ®Google Scholar
• Ax, J., Kochen, S. (1965). Diophantine problems over local fields I. Amer. J. Math. 87: 605–630. DOI: https://doi.org/10.2307/2373065.
   Google Scholar
• Baumgartner, J. (1974). A short proof of Hindman’s theorem. J. Combin. Theory Ser. A. 17(3): 384–386.
   Google Scholar
• Csikvári, P., Gyarmati, K., Sárközy, A. (2012). Density and Ramsey type results on algebraic equations with restricted solution sets. Combinatorica. 32(4): 425–449. DOI: https://doi.org/10.1007/s00493-012-2697-9.
   Web of Science ®Google Scholar
• Di Nasso, M., Goldbring, I., Lupini, M. (2019). Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory. Lecture Notes in Mathematics 2239. Cham, Switzerland: Springer.
   Google Scholar
• Ellis, R.(1958). Distal transformation groups. Pacific J. Math. 8(3): 401–405.
   Google Scholar
• Erdős, P. (1973). Problems and results on combinatorial number theory. In: Srivastava, J. N., Harary, F., Rao, C. R., Rota, G.-C., and Shrikhande, S. S. A Survey of Combinatorial Theory. Amsterdam: North-Holland, pp. 117–138.
   Google Scholar
• Furstenberg, H., Weiss, B. (1978). Topological dynamics and combinatorial number theory. J. Analyse Math. 34: 61–85. DOI: https://doi.org/10.1007/BF02790008.
   Web of Science ®Google Scholar
• Graham, R. L., Rothschild, B. L. (1971). Ramsey’s theorem for n-parameter sets. Trans. Amer. Math. Soc. 159: 257–292. DOI: https://doi.org/10.2307/1996010.
   Web of Science ®Google Scholar
• Graham, R. L., Rothschild, B., Spencer, J. H. (1990). Ramsey Theory, 2nd ed. New York: Wiley.
   Google Scholar
• Green, B. J., Lindqvist, S. (2019). Monochromatic solutions to x+y=z2 . Canad. J. Math. 71(3): 579–605.
   Web of Science ®Google Scholar
• Hindman, N. (1972). The existence of certain ultrafilters on N and a conjecture of Graham and Rothschild. Proc. Amer. Math. Soc. 36(2): 341–346.
   Web of Science ®Google Scholar
• Hindman, N. (1974). Finite sums from sequences within cells of a partition of N. J. Combin. Theory Ser. A. 17(1): 1–11.
   Google Scholar
• Hindman, N. (1979). Ultrafilters and combinatorial number theory. In: Nathanson, M. B., ed. Number Theory. Carbondale 1979, Proceedings of the Southern Illinois Number Theory Conference. Berlin: Springer-Verlag, pp. 119–184.
   Google Scholar
• Hindman, N. (1987). Ultrafilters and Ramsey theory — an update. In: Steprāns, J., Watson, S., eds. Set Theory and its Applications. Ontario, Canada 1987, Proceedings of a Conference held at York University. Berlin: Springer-Verlag, pp. 97–118.
   Google Scholar
• Hindman, N., Steprāns, J., Strauss, D. (2012). Semigroups in which all strongly summable ultrafilters are sparse. New York J. Math. 18: 835–848.
   Web of Science ®Google Scholar
• Hindman, N., Strauss, D. (2012). Algebra in the Stone–Čech compactification, Second revised and extended edition. De Gruyter Textbook. Berlin: Walter de Gruyter & Co.
   Google Scholar
• Howard, P., Rubin, J. (1991). Consequences of the Axiom of Choice. Providence, RI: American Mathematical Society.
   Google Scholar
• Numakura, K. (1952). On bicompact semigroups. Math. J. Okayama Univ. 1: 99–108.
   Google Scholar
• Ramsey, F. P. (1930). On a problem of formal logic. Proc. London Math. Soc. (2). 30(4): 264–286.
   Google Scholar
• Sanders, J. H. (1968). A generalization of Schur’s theorem. Ph.D. dissertation. New Haven, CT: Yale University.
   Google Scholar
• Schoutens, H. (2010). The Use of Ultraproducts in Commutative Algebra. Lecture Notes in Mathematics 1999. Berlin: Springer.
   Google Scholar
• Schur, I. (1912). Über die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen. Sitzungsberichte der Berliner Mathematischen Gesellschaft 11: 40–50.
   Google Scholar
• Todorčević, S. (2010). Introduction to Ramsey Spaces. Annals of Mathematics Studies 104. Princeton, NJ: Princeton Univ. Press.
   Google Scholar
• van der Waerden, B. L. (1927). Beweis einer Baudetschen Vermutung. Nieuw archief voor Wiskunde 15: 212–216.
   Google Scholar