Partition regularity of system of nonlinear equations
| Autor: | Goswami, Sayan |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | In $1979$, using the theory of ultrafilters, N. Hindman proved that for every finite coloring of $\mathbb{N}$, there exists a color that contains both additive and multiplicative $IP$ sets. Later, in $1993$, V. Bergelson and N. Hindman found an elementary proof. In this article, we prove that the partial product of these two $IP$ sets also lies in the same color. As an immediate consequence, we have the equation $a+b=c\cdot d$ partition regular. This was a conjecture of P. Csikv\'{a}ri, K. Gyarmati, and A. S\'{a}rk\"{o}zy, which was solved by V. Bergelson and N. Hindman independently. In this article, we separately give a very short proof of this conjecture. Then we extend our techniques to deduce systems of nonlinear equations that are partition regular. We prove that a relatively weak version of the nonlinear Rado systems is partition regular. Comment: arXiv admin note: text overlap with arXiv:2401.10550 |
| Databáze: | arXiv |
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