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- W4301273002 abstract "We show that for every finite colouring of the natural numbers there exists $a,b >1$ such that the triple ${a,b,a^b}$ is monochromatic. We go on to show the partition regularity of a much richer class of patterns involving exponentiation. For example, as a corollary to our main theorem, we show that for every $n in mathbb{N}$ and for every finite colouring of the natural numbers, we may find a monochromatic set including the integers $x_1,ldots,x_n >1$; all products of distinct $x_i$; and all exponential compositions of distinct $x_i$ which respect the order $x_1,ldots,x_n$. In particular, for every finite colouring of the natural numbers one can find a monochromatic quadruple of the form ${ a,b,ab,a^b }$, where $a,b>1$." @default.
- W4301273002 created "2022-10-05" @default.
- W4301273002 creator A5074703575 @default.
- W4301273002 date "2016-07-28" @default.
- W4301273002 modified "2023-09-26" @default.
- W4301273002 title "Exponential Patterns in Arithmetic Ramsey Theory" @default.
- W4301273002 doi "https://doi.org/10.48550/arxiv.1607.08396" @default.
- W4301273002 hasPublicationYear "2016" @default.
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