The Two-Colour Rado Number for the Equation ax +?by =?(a + b)z

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• 作者：Swati Gupta ; J. Thulasi Rangan ; Amitabha Tripathi
• 关键词：05C55 ; 05D10 ; Schur numbers ; Rado numbers ; colouring ; monochromatic solution ; regular equation
• 刊名：Annals of Combinatorics
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：19
• 期：2
• 页码：269-291
• 全文大小：1,015 KB
• 参考文献：1.Beutelspacher, A., Brestovansky, W.: Generalized Schur numbers. In: Jungnickel, D., Vedder, K. (eds.) Combinatorial Theory, pp. 30-8. Springer, Berlin-New York (1982)
  2.Bialostocki A., Bialostocki G., Schaal D.: A zero-sum theorem. J. Combin. Theory Ser. A 101(1), 147-52 (2003)View Article MATH MathSciNet
  3.Bialostocki A., Lefmann H., Meerdink T.: On the degree of regularity of some equations. Discrete Math. 150(1-3), 49-0 (1996)View Article MATH MathSciNet
  4.Bialostocki A., Schaal D.: On a variation of Schur numbers. Graphs Combin. 16(2), 139-47 (2000)View Article MATH MathSciNet
  5.Burr, S., Loo, S.: On Rado numbers I. Preprint.
  6.Deuber, W.A.: Developments based on Rado’s dissertation “Studien zur Kombinatorik". In: Siemons, J. (ed.) Surveys in combinatorics, 1989, pp. 52-4. Cambridge Univ. Press, Cambridge (1989)
  7.Guo S., Sun Z.-W.: Determination of the two-color Rado number for \({a_1 x_1 + \cdot + a_m x_m = x_0}\) . J. Combin. Theory Ser. A 115(2), 345-53 (2008)View Article MATH MathSciNet
  8.Harborth H., Maasberg S.: Rado numbers for homogeneous second order linear recurrences -degree of partition regularity. Congr. Numer. 108, 109-18 (1995)MATH MathSciNet
  9.Harborth, H., Maasberg, S.: Rado numbers for Fibonacci sequences and a problem of S. Rabinowitz. In: Bergum, G.E. (Ed.) Applications of Fibonacci Numbers, Vol. 6, pp. 143-53. Kluwer Academic Publishers, Dordrecht (1996)
  10.Harborth H., Maasberg S.: Rado numbers for a(x +?y) =?b z . J. Combin. Theory Ser. A 80(2), 356-63 (1997)View Article MATH MathSciNet
  11.Harborth, H., Maasberg, S.: All two-color Rado numbers for a(x +?y) =?b z . Discrete Math. 197/198, 397-07 (1999)
  12.Hopkins B., Schaal D.: On Rado numbers for \({\sum_{i=1}^{m-1} {a_i x_i} = x_m}\) . Adv. Appl. Math. 35(4), 433-41 (2005)View Article MATH MathSciNet
  13.Jones S., Schaal D.: Some 2-color Rado numbers. Congr. Numer. 152, 197-99 (2001)MATH MathSciNet
  14.Jones S., Schaal D.: Two-color Rado numbers for x +?y +?c = k z . DiscreteMath. 289(1-3), 63-9 (2004)MATH MathSciNet
  15.Kosek W., Schaal D.: Rado numbers for the equation \({\sum_{i=1}^{m-1} {x_i} + c = x_m}\) for negative values of c. Adv. Appl. Math. 27(4), 805-15 (2001)View Article MATH MathSciNet
  16.Landman, B.M., Robertson, A.: Ramsey Theory on the Integers. Student Mathematical Library, Vol. 24. Amer. Math. Soc., Providence, RI (2004)
  17.Rado, R.: Verallgemeinerung eines satzes von van der waerden mit anwendungen auf ein problem der Zahlentheorie. Sonderausg. Sitzungsber. Preuss. Akad. Wiss. Phys.Math. Klasse 17, 1-0 (1933)
  18.Rado R.: Studien zur Kombinatorik. Math. Z. 36(1), 424-70 (1933)View Article MATH MathSciNet
  19.Rado, R.: Note on combinatorial analysis. Proc. London Math. Soc. (2) 48, 122-60 (1945)
  20.Robertson A., Schaal D.: Off-diagonal generalized Schur numbers. Adv. Appl. Math. 26(3), 252-57 (2001)View Article MATH MathSciNet
  21.Schaal D.: On generalized Schur numbers. Congr. Numer. 98, 178-87 (1993)MATH MathSciNet
  22.Schur, I.: Uber die Kongruenz x m +?y m = z m (mod p). Jahresber. Deutsch. Math.-Verein. 25, 114-17 (1916)
  
• 作者单位：Swati Gupta (1)
  J. Thulasi Rangan (2)
  Amitabha Tripathi (3)
  
  1. Operations Research Center, MIT, Cambridge, MA, 02139, USA
  2. Photokaaran, 63/31 Village Street, Thiruvotriyur Market, Chennai, 600019, India
  3. Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi, 110016, India
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  
• 出版者：Birkh盲user Basel
• ISSN：0219-3094

文摘

For relatively prime positive integers a and b, let \({n = \mathcal{R}(a, b)}\) denote the least positive integer such that every 2-colouring of [1, n] admits a monochromatic solution to ax + by = (a + b)z with x, y, z distinct integers. It is known that \({\mathcal{R}(a, b) \leq 4(a + b) + 1}\). We show that \({\mathcal{R}(a, b) = 4(a + b) + 1}\), except when (a, b) =?(3, 4) or (a, b) = (1, 4k) for some \({k \geq 1}\), and \({\mathcal{R}(a, b) = 4(a + b)-1}\) in these exceptional cases.