Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: How to find the least upper bound on the van der Waerden Number $W(r,
  k)$ that is some integer Power of the coloring Integer $r$   What is a least integer upper bound on van der Waerden number $W(r, k)$ among
the powers of the integer $r$? We show how this can be found by expanding the
integer $W(r, k)$ into powers of $r$. Doing this enables us to find both a
least upper bound and a greatest lower bound on $W(r, k)$ that are some powers
of $r$ and where the greatest lower bound is equal to or smaller than $W(r,
k)$. A finite series expansion of each $W(r, k)$ into integer powers of $r$
then helps us to find also a greatest real lower bound on any $k$ for which a
conjecture posed by R. Graham is true, following immediately as a particular
case of the overall result.

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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex