Monochromatic sums and products in $\mathbb N$

Abstract

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair $\{x+y,xy\}$. We answer this question affirmatively in a strong sense by exhibiting a large new class of nonlinear patterns that can be found in a single cell of any finite partition of $\mathbb N$. Our proof involves a correspondence principle that transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as $x^2-y^2=z$ and $x^2+2y^2-3z^2=w$.

Note: To view the article, click on the URL link for the DOI number.

  • [Beiglbock11] Go to document M. Beiglböck, "An ultrafilter approach to Jin’s theorem," Israel J. Math., vol. 185, pp. 369-374, 2011.
    @ARTICLE{Beiglbock11, mrkey = {2837141},
      issn = {0021-2172},
      author = {Beiglb{ö}ck, Mathias},
      mrclass = {11B05 (03E05 03H05 28E05 43A07)},
      doi = {10.1007/s11856-011-0114-5},
      journal = {Israel J. Math.},
      zblnumber = {1300.11015},
      volume = {185},
      mrnumber = {2837141},
      fjournal = {Israel Journal of Mathematics},
      mrreviewer = {{\c{S}}. A. Basarab},
      coden = {ISJMAP},
      title = {An ultrafilter approach to {J}in's theorem},
      year = {2011},
      pages = {369--374},
      }
  • [Beiglbock_Bergelson_Hindman_Strauss06] Go to document M. Beiglböck, V. Bergelson, N. Hindman, and D. Strauss, "Multiplicative structures in additively large sets," J. Combin. Theory Ser. A, vol. 113, iss. 7, pp. 1219-1242, 2006.
    @ARTICLE{Beiglbock_Bergelson_Hindman_Strauss06, mrkey = {2259058},
      number = {7},
      issn = {0097-3165},
      author = {Beiglb{ö}ck, Mathias and Bergelson, Vitaly and Hindman, Neil and Strauss, Dona},
      mrclass = {05D10 (11B75 37A45)},
      doi = {10.1016/j.jcta.2005.11.003},
      journal = {J. Combin. Theory Ser. A},
      zblnumber = {1105.05071},
      volume = {113},
      mrnumber = {2259058},
      fjournal = {Journal of Combinatorial Theory. Series A},
      mrreviewer = {Alexander Fish},
      coden = {JCBTA7},
      title = {Multiplicative structures in additively large sets},
      year = {2006},
      pages = {1219--1242},
      }
  • [Beiglbock_Bergelson_Hindman_Strauss08] Go to document M. Beiglböck, V. Bergelson, N. Hindman, and D. Strauss, "Some new results in multiplicative and additive Ramsey theory," Trans. Amer. Math. Soc., vol. 360, iss. 2, pp. 819-847, 2008.
    @ARTICLE{Beiglbock_Bergelson_Hindman_Strauss08, mrkey = {2346473},
      number = {2},
      issn = {0002-9947},
      author = {Beiglb{ö}ck, Mathias and Bergelson, Vitaly and Hindman, Neil and Strauss, Dona},
      mrclass = {05D10},
      doi = {10.1090/S0002-9947-07-04370-X},
      journal = {Trans. Amer. Math. Soc.},
      zblnumber = {1136.05074},
      volume = {360},
      mrnumber = {2346473},
      fjournal = {Transactions of the American Mathematical Society},
      mrreviewer = {J. Spencer},
      coden = {TAMTAM},
      title = {Some new results in multiplicative and additive {R}amsey theory},
      year = {2008},
      pages = {819--847},
      }
  • [Bergelson87b] Go to document V. Bergelson, "Ergodic Ramsey theory," in Logic and Combinatorics, Providence, RI: Amer. Math. Soc., 1987, vol. 65, pp. 63-87.
    @INCOLLECTION{Bergelson87b, mrkey = {0891243},
      author = {Bergelson, Vitaly},
      mrclass = {05A05 (04A20 05C15 11B05 11K55 28D99)},
      series = {Contemp. Math.},
      address = {Providence, RI},
      publisher = {Amer. Math. Soc.},
      doi = {10.1090/conm/065/891243},
      zblnumber = {0642.10052},
      volume = {65},
      mrnumber = {0891243},
      booktitle = {Logic and Combinatorics},
      mrreviewer = {Jaroslav Ne{š}et{\v{r}}il},
      venue = {{A}rcata, {C}alif., 1985},
      title = {Ergodic {R}amsey theory},
      pages = {63--87},
      year = {1987},
      }
  • [Bergelson96] Go to document V. Bergelson, "Ergodic Ramsey theory—an update," in Ergodic Theory of ${\bf Z}^d$ Actions, Cambridge: Cambridge Univ. Press, 1996, vol. 228, pp. 1-61.
    @INCOLLECTION{Bergelson96, mrkey = {1411215},
      author = {Bergelson, Vitaly},
      mrclass = {28D05 (05A18 05D10 11B25)},
      series = {London Math. Soc. Lecture Note Ser.},
      address = {Cambridge},
      publisher = {Cambridge Univ. Press},
      doi = {10.1017/CBO9780511662812.002},
      zblnumber = {0846.05095},
      volume = {228},
      mrnumber = {1411215},
      booktitle = {Ergodic Theory of {${\bf Z}\sp d$} Actions},
      mrreviewer = {Karl Petersen},
      venue = {{W}arwick, 1993--1994},
      title = {Ergodic {R}amsey theory---an update},
      pages = {1--61},
      year = {1996},
      }
  • [Bergelson05] Go to document V. Bergelson, "Multiplicatively large sets and ergodic Ramsey theory," Israel J. Math., vol. 148, pp. 23-40, 2005.
    @ARTICLE{Bergelson05, mrkey = {2191223},
      issn = {0021-2172},
      author = {Bergelson, Vitaly},
      mrclass = {37A45 (05D10)},
      doi = {10.1007/BF02775431},
      journal = {Israel J. Math.},
      zblnumber = {1093.11003},
      volume = {148},
      mrnumber = {2191223},
      fjournal = {Israel Journal of Mathematics},
      mrreviewer = {Mathias Beiglb{ö}ck},
      coden = {ISJMAP},
      title = {Multiplicatively large sets and ergodic {R}amsey theory},
      year = {2005},
      pages = {23--40},
      }
  • [Bergelson_Furstenberg_McCutcheon96] Go to document V. Bergelson, H. Furstenberg, and R. McCutcheon, "IP-sets and polynomial recurrence," Ergodic Theory Dynam. Systems, vol. 16, iss. 5, pp. 963-974, 1996.
    @ARTICLE{Bergelson_Furstenberg_McCutcheon96, mrkey = {1417769},
      number = {5},
      issn = {0143-3857},
      author = {Bergelson, Vitaly and Furstenberg, Hillel and McCutcheon, Randall},
      mrclass = {28D15 (05D10)},
      doi = {10.1017/S0143385700010130},
      journal = {Ergodic Theory Dynam. Systems},
      zblnumber = {0958.28013},
      volume = {16},
      mrnumber = {1417769},
      fjournal = {Ergodic Theory and Dynamical Systems},
      mrreviewer = {Thomas Ward},
      title = {I{P}-sets and polynomial recurrence},
      year = {1996},
      pages = {963--974},
      }
  • [Bergelson_Hindman01] Go to document V. Bergelson and N. Hindman, "Partition regular structures contained in large sets are abundant," J. Combin. Theory Ser. A, vol. 93, iss. 1, pp. 18-36, 2001.
    @ARTICLE{Bergelson_Hindman01, mrkey = {1807110},
      number = {1},
      issn = {0097-3165},
      author = {Bergelson, Vitaly and Hindman, Neil},
      mrclass = {05D10},
      doi = {10.1006/jcta.2000.3061},
      journal = {J. Combin. Theory Ser. A},
      zblnumber = {0966.05078},
      volume = {93},
      mrnumber = {1807110},
      fjournal = {Journal of Combinatorial Theory. Series A},
      title = {Partition regular structures contained in large sets are abundant},
      year = {2001},
      pages = {18--36},
      }
  • [Bergelson_Johnson_Moreira15] V. Bergelson, J. Johnson, and J. Moreira, New polynomial and multidimensional extensions of classical partition results, 2015.
    @MISC{Bergelson_Johnson_Moreira15,
      author = {Bergelson, Vitaly and Johnson, J. and Moreira, J.},
      arxiv = {1501.02408},
      title = {New polynomial and multidimensional extensions of classical partition results},
      year = {2015},
      }
  • [Bergelson_Leibman96] Go to document V. Bergelson and A. Leibman, "Polynomial extensions of van der Waerden’s and Szemerédi’s theorems," J. Amer. Math. Soc., vol. 9, iss. 3, pp. 725-753, 1996.
    @ARTICLE{Bergelson_Leibman96, mrkey = {1325795},
      number = {3},
      issn = {0894-0347},
      author = {Bergelson, V. and Leibman, A.},
      mrclass = {11B25 (05D10 28D05 54H20)},
      doi = {10.1090/S0894-0347-96-00194-4},
      journal = {J. Amer. Math. Soc.},
      zblnumber = {0870.11015},
      volume = {9},
      mrnumber = {1325795},
      fjournal = {Journal of the American Mathematical Society},
      mrreviewer = {Pierre Michel},
      title = {Polynomial extensions of van der {W}aerden's and {S}zemerédi's theorems},
      year = {1996},
      pages = {725--753},
      }
  • [Bergelson_Leibman99] Go to document V. Bergelson and A. Leibman, "Set-polynomials and polynomial extension of the Hales-Jewett theorem," Ann. of Math., vol. 150, iss. 1, pp. 33-75, 1999.
    @ARTICLE{Bergelson_Leibman99, mrkey = {1715320},
      number = {1},
      issn = {0003-486X},
      author = {Bergelson, V. and Leibman, A.},
      mrclass = {05D10 (05A18 11B25 37B10 54H20)},
      doi = {10.2307/121097},
      journal = {Ann. of Math.},
      zblnumber = {0969.05061},
      volume = {150},
      mrnumber = {1715320},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {N. Hindman},
      coden = {ANMAAH},
      title = {Set-polynomials and polynomial extension of the {H}ales-{J}ewett theorem},
      year = {1999},
      pages = {33--75},
      }
  • [Bergelson_Leibman_Lesigne08] Go to document V. Bergelson, A. Leibman, and E. Lesigne, "Intersective polynomials and the polynomial Szemerédi theorem," Adv. Math., vol. 219, iss. 1, pp. 369-388, 2008.
    @ARTICLE{Bergelson_Leibman_Lesigne08, mrkey = {2435427},
      number = {1},
      issn = {0001-8708},
      author = {Bergelson, V. and Leibman, A. and Lesigne, E.},
      mrclass = {37A45 (11B25 11C08)},
      doi = {10.1016/j.aim.2008.05.008},
      journal = {Adv. Math.},
      zblnumber = {1156.11007},
      volume = {219},
      mrnumber = {2435427},
      fjournal = {Advances in Mathematics},
      mrreviewer = {Bryna Kra},
      coden = {ADMTA4},
      title = {Intersective polynomials and the polynomial {S}zemerédi theorem},
      year = {2008},
      pages = {369--388},
      }
  • [Bergelson_Moreira15] Go to document V. Bergelson and J. Moreira, Ergodic theorem involving additive and multiplicative groups of a field and $\{x+y,xy\}$ patterns, 2016.
    @misc{Bergelson_Moreira15,
      author = {Bergelson, Vitaly and Moreira, Joel},
      TITLE={Ergodic theorem involving additive and multiplicative groups of a field and $\{x+y,xy\}$ patterns},
      NOTE={to appear in \emph{Ergodic Theory Dynam. Systems}},
      doi = {10.1017/etds.2015.68},
      YEAR={2016},
      }
  • [Bergelson_Moreira16b] Go to document V. Bergelson and J. Moreira, "Van der Corput’s difference theorem: some modern developments," Indag. Math., vol. 27, iss. 2, pp. 437-479, 2016.
    @ARTICLE{Bergelson_Moreira16b, mrkey = {3479166},
      number = {2},
      issn = {0019-3577},
      author = {Bergelson, Vitaly and Moreira, Joel},
      mrclass = {40A05 (05-02 37A45)},
      doi = {10.1016/j.indag.2015.10.014},
      journal = {Indag. Math.},
      zblnumber = {06567937},
      volume = {27},
      mrnumber = {3479166},
      fjournal = {Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae. New Series},
      title = {Van der {C}orput's difference theorem: some modern developments},
      year = {2016},
      pages = {437--479},
      }
  • [Bergelson_Moreira16] Go to document V. Bergelson and J. Moreira, "Measure preserving actions of affine semigroups and $\{x+y,xy\}$ patterns," Ergodic Theory Dynam. Systems, p. 26, 2016.
    @ARTICLE{Bergelson_Moreira16,
      author = {Bergelson, Vitaly and Moreira, Joel},
      journal = {Ergodic Theory Dynam. Systems},
      fjournal = {Ergodic Theory and Dynamical Systems},
      title = {Measure preserving actions of affine semigroups and $\{x+y,xy\}$ patterns},
      year = {2016},
      doi = {10.1017/etds.2016.39},
      note = {published online: 26 July 2016},
      pages={ 26 pp.},
      }
  • [Cilleruelo12] Go to document J. Cilleruelo, "Combinatorial problems in finite fields and Sidon sets," Combinatorica, vol. 32, iss. 5, pp. 497-511, 2012.
    @ARTICLE{Cilleruelo12, mrkey = {3004806},
      number = {5},
      issn = {0209-9683},
      author = {Cilleruelo, Javier},
      mrclass = {11B83 (11B30)},
      doi = {10.1007/s00493-012-2819-4},
      journal = {Combinatorica},
      zblnumber = {1291.11025},
      volume = {32},
      mrnumber = {3004806},
      fjournal = {Combinatorica. An International Journal on Combinatorics and the Theory of Computing},
      mrreviewer = {Nikolai Volodin},
      title = {Combinatorial problems in finite fields and {S}idon sets},
      year = {2012},
      pages = {497--511},
      }
  • [Csikvari_Gyarmati_Sarkozy12] Go to document P. Csikvári, K. Gyarmati, and A. Sárközy, "Density and Ramsey type results on algebraic equations with restricted solution sets," Combinatorica, vol. 32, iss. 4, pp. 425-449, 2012.
    @ARTICLE{Csikvari_Gyarmati_Sarkozy12, mrkey = {2965285},
      number = {4},
      issn = {0209-9683},
      author = {Csikv{á}ri, P{é}ter and Gyarmati, Katalin and S{á}rk{ö}zy, Andr{á}s},
      mrclass = {11B75 (05D10)},
      doi = {10.1007/s00493-012-2697-9},
      journal = {Combinatorica},
      zblnumber = {1286.11026},
      volume = {32},
      mrnumber = {2965285},
      fjournal = {Combinatorica. An International Journal on Combinatorics and the Theory of Computing},
      mrreviewer = {Nikolai Volodin},
      title = {Density and {R}amsey type results on algebraic equations with restricted solution sets},
      year = {2012},
      pages = {425--449},
      }
  • [Frantzikinakis_Host14] Go to document N. Frantzikinakis and B. Host, "Higher order Fourier analysis of multiplicative functions and applications," J. Amer. Math. Soc., vol. 30, iss. 1, pp. 67-157, 2017.
    @ARTICLE{Frantzikinakis_Host14, mrkey = {3556289},
      number = {1},
      issn = {0894-0347},
      author = {Frantzikinakis, Nikos and Host, Bernard},
      mrclass = {11N37 (05D10 11B30 11N60 37A45)},
      doi = {10.1090/jams/857},
      journal = {J. Amer. Math. Soc.},
      zblnumber = {06640527},
      volume = {30},
      mrnumber = {3556289},
      fjournal = {Journal of the American Mathematical Society},
      title = {Higher order {F}ourier analysis of multiplicative functions and applications},
      year = {2017},
      pages = {67--157},
      }
  • [Furstenberg77] Go to document H. Furstenberg, "Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions," J. Analyse Math., vol. 31, pp. 204-256, 1977.
    @ARTICLE{Furstenberg77, mrkey = {0498471},
      issn = {0021-7670},
      author = {Furstenberg, Harry},
      mrclass = {10L10 (10K10 28A65)},
      doi = {10.1007/BF02813304},
      journal = {J. Analyse Math.},
      zblnumber = {0347.28016},
      volume = {31},
      mrnumber = {0498471},
      fjournal = {Journal d'Analyse Mathématique},
      mrreviewer = {Francois Aribaud},
      title = {Ergodic behavior of diagonal measures and a theorem of {S}zemerédi on arithmetic progressions},
      year = {1977},
      pages = {204--256},
      }
  • [Furstenberg81] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton, N.J.: Princeton Univ. Press, 1981.
    @BOOK{Furstenberg81, mrkey = {0603625},
      author = {Furstenberg, H.},
      mrclass = {28D05 (10K10 10L10 54H20)},
      isbn = {0-691-08269-3},
      address = {Princeton, N.J.},
      publisher = {Princeton Univ. Press},
      zblnumber = {0459.28023},
      mrnumber = {0603625},
      mrreviewer = {Michael Keane},
      title = {Recurrence in Ergodic Theory and Combinatorial Number Theory},
      year = {1981},
      pages = {xi+203},
      }
  • [Furstenberg_Weiss78] Go to document H. Furstenberg and B. Weiss, "Topological dynamics and combinatorial number theory," J. Analyse Math., vol. 34, pp. 61-85 (1979), 1978.
    @ARTICLE{Furstenberg_Weiss78, mrkey = {0531271},
      issn = {0021-7670},
      author = {Furstenberg, H. and Weiss, B.},
      mrclass = {05A17 (10A45 54H20)},
      doi = {10.1007/BF02790008},
      journal = {J. Analyse Math.},
      zblnumber = {0425.54023},
      volume = {34},
      mrnumber = {0531271},
      fjournal = {Journal d'Analyse Mathématique},
      mrreviewer = {N. Hindman},
      coden = {JOAMAV},
      title = {Topological dynamics and combinatorial number theory},
      year = {1978},
      pages = {61--85 (1979)},
      }
  • [Graham_Rothschild_Spencer80] R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey Theory, Second ed., New York: John Wiley & Sons, 1990.
    @BOOK{Graham_Rothschild_Spencer80, mrkey = {1044995},
      author = {Graham, Ronald L. and Rothschild, Bruce L. and Spencer, Joel H.},
      mrclass = {05-02 (04A20 05A99 05C55 54H20)},
      series = {Wiley-Intersci. Ser. Discrete Math. Optim.},
      edition = {Second},
      isbn = {0-471-50046-1},
      address = {New York},
      publisher = {John Wiley \& Sons},
      zblnumber = {0705.05061},
      mrnumber = {1044995},
      title = {Ramsey Theory},
      year = {1990},
      pages = {xii+196},
      }
  • [Green_Sanders15] Go to document B. Green and T. Sanders, "Monochromatic sums and products," Discrete Anal., p. 613, 2016.
    @ARTICLE{Green_Sanders15, mrkey = {3533304},
      issn = {2397-3129},
      author = {Green, Ben and Sanders, Tom},
      mrclass = {11B75 (05D05)},
      doi = {10.19086/da.613},
      journal = {Discrete Anal.},
      zblnumber = {06637038},
      mrnumber = {3533304},
      fjournal = {Discrete Analysis},
      title = {Monochromatic sums and products},
      year = {2016},
      pages = {Paper No. 613, 43},
      }
  • [Hanson13] Go to document B. Hanson, "Capturing forms in dense subsets of finite fields," Acta Arith., vol. 160, iss. 3, pp. 277-284, 2013.
    @ARTICLE{Hanson13, mrkey = {3106098},
      number = {3},
      issn = {0065-1036},
      author = {Hanson, Brandon},
      mrclass = {11B30 (05D10 11D09)},
      doi = {10.4064/aa160-3-4},
      journal = {Acta Arith.},
      zblnumber = {1316.11011},
      volume = {160},
      mrnumber = {3106098},
      fjournal = {Acta Arithmetica},
      mrreviewer = {Imre Z. Ruzsa},
      title = {Capturing forms in dense subsets of finite fields},
      year = {2013},
      pages = {277--284},
      }
  • [Hindman79] Go to document N. Hindman, "Partitions and sums and products of integers," Trans. Amer. Math. Soc., vol. 247, pp. 227-245, 1979.
    @ARTICLE{Hindman79, mrkey = {0517693},
      issn = {0002-9947},
      author = {Hindman, Neil},
      mrclass = {10A45 (05A17 54A25)},
      doi = {10.2307/1998782},
      journal = {Trans. Amer. Math. Soc.},
      zblnumber = {0401.05012},
      volume = {247},
      mrnumber = {0517693},
      fjournal = {Transactions of the American Mathematical Society},
      mrreviewer = {G. L. Alexanderson},
      coden = {TAMTAM},
      title = {Partitions and sums and products of integers},
      year = {1979},
      pages = {227--245},
      }
  • [Hindman_Imre_Strauss03] Go to document N. Hindman, I. Leader, and D. Strauss, "Open problems in partition regularity," Combin. Probab. Comput., vol. 12, iss. 5-6, pp. 571-583, 2003.
    @ARTICLE{Hindman_Imre_Strauss03, mrkey = {2037071},
      number = {5-6},
      issn = {0963-5483},
      author = {Hindman, Neil and Leader, Imre and Strauss, Dona},
      mrclass = {05D10},
      doi = {10.1017/S0963548303005716},
      journal = {Combin. Probab. Comput.},
      zblnumber = {1061.05095},
      volume = {12},
      mrnumber = {2037071},
      note = {Special issue on {R}amsey theory},
      fjournal = {Combinatorics, Probability and Computing},
      mrreviewer = {Bruce Landman},
      title = {Open problems in partition regularity},
      year = {2003},
      pages = {571--583},
      }
  • [Hindman_Strauss98] Go to document N. Hindman and D. Strauss, Algebra in the Stone-\v Cech Compactification, Berlin: Walter de Gruyter & Co., 1998, vol. 27.
    @BOOK{Hindman_Strauss98, mrkey = {1642231},
      author = {Hindman, Neil and Strauss, Dona},
      mrclass = {54-02 (05D10 22A15 54D35 54H99)},
      series = {de Gruyter Exp. Math.},
      isbn = {3-11-015420-X},
      address = {Berlin},
      publisher = {Walter de Gruyter \& Co.},
      doi = {10.1515/9783110809220},
      zblnumber = {0918.22001},
      volume = {27},
      mrnumber = {1642231},
      mrreviewer = {Michael Tka{\v{c}}enko},
      title = {Algebra in the {S}tone-\v {C}ech Compactification},
      year = {1998},
      pages = {xiv+485},
      }
  • [McCutcheon10] Go to document R. McCutcheon, "A variant of the density Hales-Jewett theorem," Bull. Lond. Math. Soc., vol. 42, iss. 6, pp. 974-980, 2010.
    @ARTICLE{McCutcheon10, mrkey = {2740017},
      number = {6},
      issn = {0024-6093},
      author = {McCutcheon, R.},
      mrclass = {11B25 (05D10)},
      doi = {10.1112/blms/bdq051},
      journal = {Bull. Lond. Math. Soc.},
      zblnumber = {1262.05150},
      volume = {42},
      mrnumber = {2740017},
      fjournal = {Bulletin of the London Mathematical Society},
      mrreviewer = {T. C. Brown},
      title = {A variant of the density {H}ales-{J}ewett theorem},
      year = {2010},
      pages = {974--980},
      }
  • [Rado33] Go to document R. Rado, "Studien zur kombinatorik," Math. Zeit., vol. 36, p. 424, 1933.
    @ARTICLE{Rado33, zblnumber = {0006.14603},
      volume = {36},
      author = {Rado, R.},
      title = {Studien zur kombinatorik},
      year = {1933},
      pages = {424–470},
      journal = {Math. Zeit.},
      doi = {10.1007/BF01188632},
      mrnumber = {1545354},
      }
  • [Sarkozy78] Go to document A. SárkHozy, "On difference sets of sequences of integers. I," Acta Math. Acad. Sci. Hungar., vol. 31, iss. 1–2, pp. 125-149, 1978.
    @ARTICLE{Sarkozy78, mrkey = {0466059},
      number = {1--2},
      issn = {0001-5954},
      author = {S{á}rk{ő}zy, A.},
      mrclass = {10L10 (10J10)},
      doi = {10.1007/BF01896079},
      journal = {Acta Math. Acad. Sci. Hungar.},
      zblnumber = {0387.10033},
      volume = {31},
      mrnumber = {0466059},
      fjournal = {Acta Mathematica Academiae Scientiarum Hungaricae},
      mrreviewer = {S. L. G. Choi},
      title = {On difference sets of sequences of integers. {I}},
      year = {1978},
      pages = {125--149},
      }
  • [wtf] Go to document I. Schur, "Über die kongruenz $x^m+y^m\equiv z^m(\text{mod } p)$," Jahresbericht der Deutschen Math. Verein., vol. 25, pp. 114-117, 1916.
    @ARTICLE{wtf, zblnumber = {46.0193.02},
      volume = {25},
      author = {Schur, I.},
      title = {{Ü}ber die kongruenz $x^m+y^m\equiv z^m(\text{mod } p)$},
      year = {1916},
      pages = {114--117},
      journal = {Jahresbericht der Deutschen Math. Verein.},
      url = {http://eudml.org/doc/145475},
      }
  • [Shkredov10] Go to document I. D. Shkredov, "On monochromatic solutions of some nonlinear equations in $\Bbb Z/p\Bbb Z$," Mat. Zametki, vol. 88, iss. 4, pp. 625-634, 2010.
    @ARTICLE{Shkredov10, mrkey = {2882224},
      number = {4},
      issn = {0025-567X},
      author = {Shkredov, I. D.},
      mrclass = {11B75 (05D05 05D10 11B30 11T30)},
      doi = {10.1134/S0001434610090336},
      journal = {Mat. Zametki},
      zblnumber = {1232.05225},
      volume = {88},
      mrnumber = {2882224},
      fjournal = {Rossiĭskaya Akademiya Nauk. Matematicheskie Zametki},
      mrreviewer = {Serge{\u\i} V. Konyagin},
      title = {On monochromatic solutions of some nonlinear equations in {$\Bbb Z/p\Bbb Z$}},
      year = {2010},
      pages = {625--634},
      }
  • [Szemeredi75] E. Szemerédi, "On sets of integers containing no $k$ elements in arithmetic progression," Acta Arith., vol. 27, pp. 199-245, 1975.
    @ARTICLE{Szemeredi75, mrkey = {0369312},
      issn = {0065-1036},
      author = {Szemer{é}di, E.},
      mrclass = {10L10},
      journal = {Acta Arith.},
      zblnumber = {0303.10056},
      volume = {27},
      mrnumber = {0369312},
      fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      mrreviewer = {S. L. G. Choi},
      title = {On sets of integers containing no {$k$} elements in arithmetic progression},
      year = {1975},
      pages = {199--245},
      }
  • [vdWaerden27] B. L. van der Waerden, "Beweis einer baudetschen vermutung," Nieuw. Arch. Wisk., vol. 15, pp. 212-216, 1927.
    @ARTICLE{vdWaerden27, zblnumber = {53.0073.12},
      volume = {15},
      author = {van~der Waerden, B. L.},
      title = {Beweis einer baudetschen vermutung},
      year = {1927},
      pages = {212--216},
      journal = {Nieuw. Arch. Wisk.},
      }
  • [Vinh14] Go to document L. A. Vinh, "Monochromatic sum and product in $\Bbb{Z}/m\Bbb{Z}$," J. Number Theory, vol. 143, pp. 162-169, 2014.
    @ARTICLE{Vinh14, mrkey = {3227340},
      issn = {0022-314X},
      author = {Vinh, Le Anh},
      mrclass = {11B75},
      doi = {10.1016/j.jnt.2014.02.004},
      journal = {J. Number Theory},
      zblnumber = {1296.11017},
      volume = {143},
      mrnumber = {3227340},
      fjournal = {Journal of Number Theory},
      mrreviewer = {Donald Jason Gibson},
      title = {Monochromatic sum and product in {$\Bbb{Z}/m\Bbb{Z}$}},
      year = {2014},
      pages = {162--169},
      }

Authors

Joel Moreira

Northwestern University, Evanston, IL