Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exponential Patterns in Arithmetic Ramsey Theory   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 $a,b>1$.

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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph