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Regularity of certain diophantine equations

  • Bidisha Roy
  • Subha SarkarEmail author
Article
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Abstract

In Ramsey theory, there is a vast literature on regularity questions of linear diophantine equations. Some problems in higher degree have been considered recently. Here, we show that, for every pair of positive integers r and n, there exists an integer \(B=B(r)\) such that the diophantine equation
$$\begin{aligned}&\prod _{m=1}^{n}\left( \sum _{i=1}^{k_m} a_{m,i} x_{m,i} - \sum _{j=1}^{l_m}b_{m,j}y_{m,j}\right) = B \end{aligned}$$
with
$$\begin{aligned}&\sum _{i=1}^{k_m} a_{m,i} = \sum _{j=1}^{l_m}b_{m,j} \qquad \forall m = 1,\ldots , n \end{aligned}$$
is r-regular, where \( k_m\), \(l_m\) are also positive integers and \(a_{m,i}, b_{m,j}\) are non-zero integers.

Keywords

Monochromatic solution arithmetic progression Szemerédi theorem 

1991 Mathematics Subject Classification

05D10 11B25 11B30 11D72 
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Notes

Acknowledgements

The authors would like to sincerely thank Prof. S D Adhikari for his insightful remarks and for having fruitful discussions with him. They thank the referee for going through the manuscript meticulously and giving suggestions to improve the presentation of the paper. They are also grateful to the Department of Atomic Energy, Government of India and Harish-Chandra Research Institute for providing financial support to carry out this research.

References

  1. 1.
    Adhikari S D, Boza L, Eliahou S, Revuelta M P and Sanz M I, On the exact degree of regularity of a certain quadratic diophantine equation, European J. Combinatorics 70 (2018) 50–60MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adhikari S D and Eliahou S, On a conjecture of Fox and Kleitman on the degree of regularity of a certain linear equation, Combinatorial and Additive Number Theory II: CANT New York, NY USA, 2015 and 2016 Springer, New York (2017), Springer Proceedings in Mathematics Statistics Series, volume 220Google Scholar
  3. 3.
    Adhikari S D, Boza L, Eliahou S, Revuelta M P and Sanz M I, Equation-regular sets and the Fox–Kleitman conjecture, Discrete Math. 341 (2018) 287–298Google Scholar
  4. 4.
    Alexeev B, Fox J and Graham R, On minimal colorings without monochromatic solutions to a linear equation, Integers: Electronic. J. Combinatorial Number Theory 7(2) (2007) p. A01MathSciNetzbMATHGoogle Scholar
  5. 5.
    Alexeev B and Tsimerman J, Equations resolving a conjecture of rado on partition regularity, J. Combinatorial Theory, Ser. A 117 (2010) 1008–1010Google Scholar
  6. 6.
    Fox J and Kleitman D J, On Rado’s boundedness conjecture, J. Combin. Theory Ser. A 113 (2006) 84–100MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fox J and Radoičić R, On the degree of regularity of generalized van der Waerden triples, Integers: Electronic J. Combinatorial Number Theory 5(1) (2005) A32MathSciNetzbMATHGoogle Scholar
  8. 8.
    Golowich N, Resolving a conjecture on degree of regularity of linear homogeneous equations, The Electronic J. Combinatorics, 21(3) (2014) p3.28Google Scholar
  9. 9.
    Graham R L, Rothschild B L and Spencer J H, Ramsey Theory, 2nd edition (1990) (Wiley)Google Scholar
  10. 10.
    Rado R, Studien zur Kombinatorik, Math. Z. 36 (1933) 424–480MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rado R, Some recent results in combinatorial analysis, Congress International deas mathematiciens (1936) (Osla)Google Scholar
  12. 12.
    Schoen T and Taczala K, The degree of regularity of the equation \(\sum _{i=1}^n x_i = \sum _{i=1}^n y_i +b\), Moscow J. Combin. Number Theory 7 (2017) 74–93zbMATHGoogle Scholar
  13. 13.
    Szemerédi E, On sets of integers containing no \(k\) elements in arithmetic progression, Acta Arithmetica, 27 (1975) 199–245MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Harish-Chandra Research InstituteJhunsiIndia

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