## 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 with is

*r*and*n*, there exists an integer \(B=B(r)\) such that the diophantine equation$$\begin{aligned}&\prod _{m=1}^{n}\left( \sum _{i=1}^{k_m} a_{m,i} x_{m,i} - \sum _{j=1}^{l_m}b_{m,j}y_{m,j}\right) = B \end{aligned}$$

$$\begin{aligned}&\sum _{i=1}^{k_m} a_{m,i} = \sum _{j=1}^{l_m}b_{m,j} \qquad \forall m = 1,\ldots , n \end{aligned}$$

*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## 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.

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