Abstract
Let p be a prime, \(n\in {\mathbb {Z}^+}\) and \(w\in (0,1)\). Given a colouring \(\chi :{\mathbb {Z}/p\mathbb {Z}}\rightarrow \{1,2,\ldots ,n\}\) and a linear equation
with \(a_1,a_2,\ldots ,a_n\in {(\mathbb {Z}/p\mathbb {Z}})^{*}\) and \(b\in {\mathbb {Z}/p\mathbb {Z}}\), we denote by \(R(\chi ,{\mathcal {L}})\) the family of vectors \((b_1,b_2,\ldots ,b_n)\in ({\mathbb {Z}/p\mathbb {Z}})^n\) suchthat \(a_1b_1+a_2b_2+\cdots +a_nb_n=b\) ad \(\chi ^{-1}(i)\cap \{b_1,b_2,\ldots ,b_n\}\ne \emptyset \) for each \(i\in \{1,2,\ldots ,n\}\). In this paper it is shown that there exists a constant \(c=c(w,n)>0\) with the following property: if \(\min _{1\le i\le n}|\chi ^{-1}(i)|\ge wp+1\gg p^{\frac{3}{4}}\) and if there exist coefficients \(a_i\) and \(a_j\) such that \(a_i\not \in \{\pm a_j\}\), then
Moreover, this statement is sharp in different directions. A result about the solutions of \({\mathcal {L}}\) in a grid is used in its proof and it is interesting in its own right.
Similar content being viewed by others
Data Availibility Statement
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Axenovich, M., Fon Der Flaass, D.: On rainbow arithmetic progressions. Electron. J. Combin. 11, R1 (2004)
Balandraud, E.: Coloured solutions of equations in finite groups. J. Combin. Theory Ser. A 114(5), 854–866 (2007)
Bergelson, V., Deuber, W.A., Hindman,H.: Rado’s theorem for finite fields, in: Sets, Graphs and Numbers (Budapest, 1991), in: Colloq. Math. Soc. János Bolyai, vol. 60, 77–88 (1992)
Cameron, P.J., Cilleruelo, J., Serra, O.: On monochromatic solutions of equations in groups. Rev. Mat. Iberoam. 23(1), 385–395 (2007)
Conlon, D.: Rainbow solutions of linear equations over \(\mathbb{Z} _p\). Discrete Math. 306, 2056–2063 (2006)
Conlon, D., Jungić, V., Radoičić, R.: On the existence of rainbow 4-term arithmetic progressions. Graphs Combin. 23, 249–254 (2007)
Fox, J., Mahdian, M., Radoičić, R.: Rainbow Solutions to the Sidon Equation. Discrete Math. 308, 4773–4778 (2008)
Grynkiewicz, D.J.: Structural Additive Theory, Developments in Mathematics, vol. 30. Springer, New York (2013)
Huicochea, M., Montejano, A.: The structure of rainbow-free colorings for linear equations on three variables in \(\mathbb{Z} _p\). Integers 15A, A8 (2015)
Huicochea, M.: An inverse theorem in \(\mathbb{F} _p\) and rainbow-free colorings. Mosc. J. Comb. Number Theory 7(2), 22–58 (2017)
Huicochea, M.: Almost arithmetic progressions in \(\mathbb{F} _p\). Notes Number Theory Discrete Math. 24(1), 61–75 (2018)
Huicochea, M.: On the number of rainbow solutions of linear equations in \({\mathbb{Z} /p\mathbb{Z} }\). Aust. J. Combin. 78, 118–132 (2020)
Jungić, V., Licht, J., Mahdian, M., Nešetřil, J., Radoičić, R.: Rainbow Ramsey Theory and Anti-Ramsey results. Combin. Probab. Comput. 12, 599–620 (2005)
Jungić, V., Nešetřil, J., Radoičić, R.: Rainbow Ramsey Theory. Integers 5(2), A9 (2003)
Jungić, V., Radoičić, R.: Rainbow 3-term Arithmetic Progressions. Integers 3, A18 (2003)
Landman, B., Robertson, A.: Ramsey Theory on the Integers, Student Mathematical Library Vol. 73 AMS (2015)
Lev, V.: Restricted set addition in groups, II. A generalization of the Erdős-Heilbronn conjecture. Electron. J. Combin. 7(4), 1–10 (2000)
Llano, B., Montejano, A.: Rainbow-free colorings for \(x + y = cz\) in \(Z_p\). Discrete Math. 312, 2566–2573 (2012)
Montejano, A., Serra, O.: Rainbow-free three colorings in abelian groups. Electron. J. Combin. 19, P45 (2012)
Montejano, A., Serra, O.: Counting patterns in colored orthogonal arrays. Discrete Math. 317, 44–52 (2014)
Nathanson, M.B.: Additive Number Theory Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics, vol. 165. Springer, New York (1996)
Pollard, J.M.: A generalisation of the theorem of Cauchy and Davenport. J. Lond. Math. Soc. 8, 460–462 (1974)
Serra, O., Vena, L.: On the number of monochromatic solutions of integer linear systems on abelian groups. Eur. J. Combin. 35, 459–473 (2014)
Acknowledgements
We would like to thank the referee for his/her positive and insightful comments and advices to improve this paper.
Funding
The author declares that no funds, grants, or other support were received during the preparation of this manuscript.
Author information
Authors and Affiliations
Contributions
There was a unique author of this paper so all the content of it was written by him.
Corresponding author
Ethics declarations
Conflict of interest
The author has no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Huicochea, M. Rainbow Solutions of a Linear Equation with Coefficients in \({\mathbb {Z}/p\mathbb {Z}}\). Graphs and Combinatorics 40, 110 (2024). https://doi.org/10.1007/s00373-024-02843-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00373-024-02843-z