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Rainbow Solutions of a Linear Equation with Coefficients in \({\mathbb {Z}/p\mathbb {Z}}\)

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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

$$\begin{aligned} \mathcal {L}:\qquad a_1x_1+a_2x_2+\cdots +a_nx_n=b \end{aligned}$$

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

$$\begin{aligned} |R(\chi ,{\mathcal {L}})|\ge cp^{n-1}. \end{aligned}$$

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.

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References

  1. Axenovich, M., Fon Der Flaass, D.: On rainbow arithmetic progressions. Electron. J. Combin. 11, R1 (2004)

    Article  MathSciNet  Google Scholar 

  2. Balandraud, E.: Coloured solutions of equations in finite groups. J. Combin. Theory Ser. A 114(5), 854–866 (2007)

    Article  MathSciNet  Google Scholar 

  3. 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)

  4. Cameron, P.J., Cilleruelo, J., Serra, O.: On monochromatic solutions of equations in groups. Rev. Mat. Iberoam. 23(1), 385–395 (2007)

    Article  MathSciNet  Google Scholar 

  5. Conlon, D.: Rainbow solutions of linear equations over \(\mathbb{Z} _p\). Discrete Math. 306, 2056–2063 (2006)

    Article  MathSciNet  Google Scholar 

  6. Conlon, D., Jungić, V., Radoičić, R.: On the existence of rainbow 4-term arithmetic progressions. Graphs Combin. 23, 249–254 (2007)

    Article  MathSciNet  Google Scholar 

  7. Fox, J., Mahdian, M., Radoičić, R.: Rainbow Solutions to the Sidon Equation. Discrete Math. 308, 4773–4778 (2008)

    Article  MathSciNet  Google Scholar 

  8. Grynkiewicz, D.J.: Structural Additive Theory, Developments in Mathematics, vol. 30. Springer, New York (2013)

    Book  Google Scholar 

  9. Huicochea, M., Montejano, A.: The structure of rainbow-free colorings for linear equations on three variables in \(\mathbb{Z} _p\). Integers 15A, A8 (2015)

    Google Scholar 

  10. Huicochea, M.: An inverse theorem in \(\mathbb{F} _p\) and rainbow-free colorings. Mosc. J. Comb. Number Theory 7(2), 22–58 (2017)

    MathSciNet  Google Scholar 

  11. Huicochea, M.: Almost arithmetic progressions in \(\mathbb{F} _p\). Notes Number Theory Discrete Math. 24(1), 61–75 (2018)

    Article  Google Scholar 

  12. Huicochea, M.: On the number of rainbow solutions of linear equations in \({\mathbb{Z} /p\mathbb{Z} }\). Aust. J. Combin. 78, 118–132 (2020)

    Google Scholar 

  13. Jungić, V., Licht, J., Mahdian, M., Nešetřil, J., Radoičić, R.: Rainbow Ramsey Theory and Anti-Ramsey results. Combin. Probab. Comput. 12, 599–620 (2005)

    Article  Google Scholar 

  14. Jungić, V., Nešetřil, J., Radoičić, R.: Rainbow Ramsey Theory. Integers 5(2), A9 (2003)

    MathSciNet  Google Scholar 

  15. Jungić, V., Radoičić, R.: Rainbow 3-term Arithmetic Progressions. Integers 3, A18 (2003)

    MathSciNet  Google Scholar 

  16. Landman, B., Robertson, A.: Ramsey Theory on the Integers, Student Mathematical Library Vol. 73 AMS (2015)

  17. Lev, V.: Restricted set addition in groups, II. A generalization of the Erdős-Heilbronn conjecture. Electron. J. Combin. 7(4), 1–10 (2000)

    Google Scholar 

  18. Llano, B., Montejano, A.: Rainbow-free colorings for \(x + y = cz\) in \(Z_p\). Discrete Math. 312, 2566–2573 (2012)

    Article  MathSciNet  Google Scholar 

  19. Montejano, A., Serra, O.: Rainbow-free three colorings in abelian groups. Electron. J. Combin. 19, P45 (2012)

    Article  Google Scholar 

  20. Montejano, A., Serra, O.: Counting patterns in colored orthogonal arrays. Discrete Math. 317, 44–52 (2014)

    Article  MathSciNet  Google Scholar 

  21. Nathanson, M.B.: Additive Number Theory Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics, vol. 165. Springer, New York (1996)

    Book  Google Scholar 

  22. Pollard, J.M.: A generalisation of the theorem of Cauchy and Davenport. J. Lond. Math. Soc. 8, 460–462 (1974)

    Article  MathSciNet  Google Scholar 

  23. Serra, O., Vena, L.: On the number of monochromatic solutions of integer linear systems on abelian groups. Eur. J. Combin. 35, 459–473 (2014)

    Article  MathSciNet  Google Scholar 

Download references

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We would like to thank the referee for his/her positive and insightful comments and advices to improve this paper.

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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

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