Korean J. Math. Vol. 28 No. 2 (2020) pp.379-389
DOI: https://doi.org/10.11568/kjm.2020.28.2.379

$2$-color Rado number for $x_1 +x_2 +\cdots +x_n=y_1 +y_2=z$

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Published: Jun 30, 2020
Keywords:
Rado number, Schur number, Ramsey theory, r-coloring
Mathematics Subject Classification:
05D10, 05C55

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Byeong Moon Kim
Department of Mathematics Gangneung-Wonju National University, Gangneung 25457, Korea
Woonjae Hwang
Division of Applied Mathematical Sciences Korea University, Sejong 30019, Korea
Byung Chul Song
Department of Mathematics Gangneung-Wonju National University, Gangneung 25457, Korea

Abstract

An $r$-color Rado number $N=R(\mathcal{L},r)$ for a system $\mathcal{L}$ of equations is the least integer, provided it exists, such that for every $r$-coloring of the set $\{1,2, \dots, N\}$, there is a monochromatic solution to $\mathcal{L}$. In this paper, we study the $2$-color Rado number $R(\mathcal{E},2)$ for $\mathcal{E}: x_1 +x_2 +\cdots +x_n=y_1 +y_2=z$ when $n\ge 4$.



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Supporting Agencies

Basic Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education

References

[1] T. Ahmed and D. Schaal, On generalized Schur numbers, Exp. Math. 25 213--218(2016). Google Scholar

[2] A. Beutelspacher and W. Brestovansky, Generalized Schur number, Lecture Notes in Math. 969 30--38(1982). Google Scholar

[3] W. Deuber, Developments based on Rado's dissertation `Studien zur Kombinatorik', in: Survey Combin., Cambridge University Press, 52--74(1989). Google Scholar

[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 115 345--353(2008). Google Scholar

[5] M.J.H. Heule, Schur number five, The Thirty-Second AAAI Conference on Articial Intelligence (AAAI-18), (2018). Google Scholar

[6] H. Harborth and S. Maasberg, Rado numbers for a(x + y) = bz, J. Combin. Theory Ser. A 80 356--363(1997). Google Scholar

[7] H. Harborth and S. Maasberg, All two-color Rado numbers for a(x+y) = bz, Discrete Math. 197/198 397--407(1999). Google Scholar

[8] B. Hopkins and D. Schaal, On Rado numbers for Google Scholar

[9] sum_{i=1}^{m-1} a_ix_i = x_m, Adv. in Appl. Math. 35 433--441 (2005). Google Scholar

[10] W. Kosek and D. Schaal, A note on disjunctive Rado numbers, Adv. in Appl. Math. 31 433--439(2003). Google Scholar

[11] R. Rado, Studien zur Kombinatorik, Math. Z. 36, 424--480 (1933). Google Scholar

[12] A. Robertson and K. Myers, Some two color, four variable Rado numbers, Adv. in Appl. Math. 41 214--226(2008). Google Scholar

[13] A. Robertson and D. Schaal, Off-diagonal generalized Schur numbers, Adv. in Appl. Math. 26 252--257(2001). Google Scholar

[14] D. Saracino, The 2-color rado number of x_1 +x_2 +...+x_n = y_1 +y_2 +...+y_k, Ars Combinatoria 129, 315--321 (2016). Google Scholar

[15] D. Saracino, The 2-color rado number of x_1 + x_2 + ... + x_{m-1} = ax_m, Ars Combinatoria 113, 81--95(2014). Google Scholar

[16] D. Saracino, The 2-color rado number of x_1+x_2+...+x_{m-1} = ax_m, II, Ars Combinatoria 119, 193--210(2015). Google Scholar

[17] D. Saracino and B. Wynne, The 2-color Rado number of x+y+kz = 3w, Ars Combinatoria 90, 119--127 (2009). Google Scholar

[18] I. Schur, Uueber die Kongruenz x^m + y^m equiv z^m (mod p), Jahresber. Deutsch. Math.-Verein. 25 114--117(1916). Google Scholar

[19] W. Wallis, A. Street and J. Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Lecture Notes in Math. 48, Springer(1972). Google Scholar