Some new Ramsey families on natural numbers

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Sciences, Shahed University, Tehran, Iran

2 Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

Abstract

In this paper, we present some new Ramsey families by using the van der Waerden theorem and affine topological correspondence principle.

Keywords

[1] M. Beiglb¨ock, V. Bergelson, N. Hindman and D. Strauss Multiplicative structures in additively large sets, J. Combin. Theory Ser. A 113 (2006), no. 7, 1219—1242.
[2] M. Beiglb¨ock, V. Bergelson, N. Hindman and D. Strauss, Some new results in multiplicative and additive Ramsey theory, Trans. Amer. Math. Soc. 360 (2008), no. 2, 819—847.
[3] V. Bergelson Multiplicatively large sets and ergodic Ramsey theory, Israel J. Math. 148 (2005), 23—40.
[4] V. Bergelson, Ergodic Ramsey theory, logic and combinatorics (Arcata, Calif., 1985), Contemp. Math. Amer. Math. Soc. 67 (1987), 63—87.
[5] V. Bergelson, H. Furstenberg and R. McCutcheon, IP-sets and polynomial recurrence, Ergodic Theory Dynam. Syst. 16 (1996), no. 5, 963—974.
[6] V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden’s and Szemer´edi’s theorems, J. Amer. Math. Soc. 9 (1996), 725–753.
[7] A. Brauer, Uber Sequenzen von Potenzresten, ¨ Wiss. Berlin Kl. Math. Phys. Tech. (1928), 9–16.
[8] W. Deuber, Partitionen und lineare Gleichungssysteme, Math. Z. 133 (1973), 109—123.
[9] N. Frantzikinakis and B. Host, Higher order Fourier analysis of multiplicative functions and applications, J. Amer. Math. Soc. 30 (2017), no. 1, 67–157.
[10] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemer´edi on arithmetic progressions, J. Anal. Math. 31 (1977), 204—256.
[11] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton, N.J., 1981.
[12] N. Hindman and D. Strauss, Algebra in the Stone-Cech Compactification: Theory and Application, ˇ second edition, de Gruyter, Berlin, 2011.
[13] R. McCutcheon A variant of the density Hales-Jewett theorem, Bull. Lond. Math. Soc. 42 (2010), no. 6, 974—980.
[14] J. Moreira, Monochromatic sums and products in N, Ann. Math. 185 (2017), 1069–1090.
[15] R. Rado, Studien zur kombinatorik, Math. Zeit. 36 (1933), 424–470.
[16] A. S´ark¨ozy, On difference sets of sequences of integers. I, Acta Math. Acad. Sci. Hungar. 31 (1978), no. 1–2, 125—149.
[17] I. Schur, Uber die Kongruenz $x^m + y^m\equiv z^m(mod p)$, Jahresbericht Deutschen Math. Verein. 25 (1916), 114—117.
[18] B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw. Arch. Wisk. 15 (1927), 212—216.
Volume 14, Issue 1
January 2023
Pages 33-38
  • Receive Date: 12 October 2021
  • Revise Date: 19 November 2021
  • Accept Date: 15 December 2021