Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Effectiveness of Hindman's theorem for bounded sums   We consider the strength and effective content of restricted versions of
Hindman's Theorem in which the number of colors is specified and the length of
the sums has a specified finite bound. Let $\mathsf{HT}^{\leq n}_k$ denote the
assertion that for each $k$-coloring $c$ of $\mathbb{N}$ there is an infinite
set $X \subseteq \mathbb{N}$ such that all sums $\sum_{x \in F} x$ for $F
\subseteq X$ and $0 < |F| \leq n$ have the same color. We prove that there is a
computable $2$-coloring $c$ of $\mathbb{N}$ such that there is no infinite
computable set $X$ such that all nonempty sums of at most $2$ elements of $X$
have the same color. It follows that $\mathsf{HT}^{\leq 2}_2$ is not provable
in $\mathsf{RCA}_0$ and in fact we show that it implies $\mathsf{SRT}^2_2$ in
$\mathsf{RCA}_0$. We also show that there is a computable instance of
$\mathsf{HT}^{\leq 3}_3$ with all solutions computing $0'$. The proof of this
result shows that $\mathsf{HT}^{\leq 3}_3$ implies $\mathsf{ACA}_0$ in
$\mathsf{RCA}_0$.

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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: atom-ph
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an