Korean Journal of Mathematics

  • 제28권2호
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  • Pages.379-389
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  • 2020
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  • 1976-8605(pISSN)
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  • 2288-1433(eISSN)
강원경기수학회 (The Kangwon-Kyungki Mathematical Society)
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2-COLOR RADO NUMBER FOR x1 + x2 + ⋯ + xn = y1 + y2 = z

  • Kim, Byeong Moon (Department of Mathematics Gangneung-Wonju National University) ;
  • Hwang, Woonjae (Division of Applied Mathematical Sciences Korea University) ;
  • Song, Byung Chul (Department of Mathematics Gangneung-Wonju National University)
  • 투고 : 2020.01.19
  • 심사 : 2020.06.22
  • 발행 : 2020.06.30
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초록

An r-color Rado number N = R(𝓛, r) for a system 𝓛 of equations is the least integer, provided it exists, such that for every r-coloring of the set {1, 2, …, N}, there is a monochromatic solution to 𝓛. In this paper, we study the 2-color Rado number R(𝓔, 2) for 𝓔 : x1 + x2 + ⋯ + xn = y1 + y2 = z when n ≥ 4.

키워드

과제정보

The authors would like to thank anonymous referee for valuable comments and corrections.

참고문헌

  1. T. Ahmed and D. Schaal, On generalized Schur numbers, Exp. Math. 25 (2016), 213-218. https://doi.org/10.1080/10586458.2015.1070776
  2. A. Beutelspacher and W. Brestovansky, Generalized Schur number, Lecture Notes in Math. 969 (1982), 30-38. https://doi.org/10.1007/BFb0062985
  3. W. Deuber, Developments based on Rado's dissertation 'Studien zur Kombinatorik', in: Survey Combin., Cambridge University Press, 52-74 (1989).
  4. S. Guo and Z.-W. Sun, Determination of the two-color Rado number for $a_1x_1$ + ... + $a_mx_m$ = $x_0$, J. Combin. Theory Ser. A 115 (2008), 345-353. https://doi.org/10.1016/j.jcta.2007.06.001
  5. M.J.H. Heule, Schur number five, The Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18),(2018).
  6. H. Harborth and S. Maasberg, Rado numbers for a(x + y) = bz, J. Combin. Theory Ser. A 80 (1997), 356-363. https://doi.org/10.1006/jcta.1997.2810
  7. H. Harborth and S. Maasberg, All two-color Rado numbers for a(x + y) = bz, Discrete Math. 197/198 (1999), 397-407. https://doi.org/10.1016/S0012-365X(98)00248-9
  8. B. Hopkins and D. Schaal, On Rado numbers for ${\sum_{i=1}^{m-1}}a_ix_i$ = $x_m$, Adv. in Appl. Math. 35 (2005), 433-441. https://doi.org/10.1016/j.aam.2005.05.003
  9. W. Kosek and D. Schaal, A note on disjunctive Rado numbers, Adv. in Appl. Math. 31 (2003), 433-439. https://doi.org/10.1016/S0196-8858(03)00020-4
  10. R. Rado, Studien zur Kombinatorik, Math. Z. 36 (1933), 424-480. https://doi.org/10.1007/BF01188632
  11. A. Robertson and K. Myers, Some two color, four variable Rado numbers, Adv. in Appl. Math. 41 (2008), 214-226. https://doi.org/10.1016/j.aam.2007.06.002
  12. A. Robertson and D. Schaal, Off-diagonal generalized Schur numbers, Adv. in Appl. Math. 26 (2001), 252-257. https://doi.org/10.1006/aama.2000.0718
  13. D. Saracino, The 2-color rado number of $x_1$ + $x_2$ + ... + $x_n$ = $y_1$ + $y_2$ + ... + yk, Ars Combinatoria 129 (2016), 315-321.
  14. D. Saracino, The 2-color rado number of $x_1$ + $x_2$ + ... + $x_{m-1}$ = $ax_m$, Ars Combinatoria 113 (2014), 81-95.
  15. D. Saracino, The 2-color rado number of $x_1$ + $x_2$ + ... + $x_{m-1}$ = $ax_m$, II, Ars Combinatoria 119 (2015), 193-210.
  16. D. Saracino and B. Wynne, The 2-color Rado number of x + y + kz = 3w, Ars Combinatoria 90 (2009), 119-127.
  17. I. Schur, Uber die Kongruenz $x^m$ + $y^m$ ${\equiv}$ $z^m$ (mod p), Jahresber. Deutsch. Math.-Verein. 25 (1916), 114-117.
  18. W. Wallis, A. Street and J. Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Lecture Notes in Math. 48, Springer (1972).