Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Strong failures of higher analogs of Hindman's theorem   We show that various analogs of Hindman's Theorem fail in a strong sense when
one attempts to obtain uncountable monochromatic sets:
  Theorem 1: There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such
that for every $X\subseteq\mathbb R$ with $|X|=|\mathbb R|$, and every colour
$\gamma\in\mathbb Q$, there are two distinct elements $x_0,x_1$ of $X$ for
which $c(x_0+x_1)=\gamma$. This forms a simultaneous generalization of a
theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah.
  Theorem 2: For every Abelian group $G$, there exists a colouring
$c:G\rightarrow\mathbb Q$ such that for every uncountable $X\subseteq G$, and
every colour $\gamma$, for some large enough integer $n$, there are pairwise
distinct elements $x_0,\ldots,x_n$ of $X$ such that $c(x_0+\cdots+x_n)=\gamma$.
In addition, it is consistent that the preceding statement remains valid even
after enlarging the set of colours from $\mathbb Q$ to $\mathbb R$.
  Theorem 3: Let $\circledast_\kappa$ assert that for every Abelian group $G$
of cardinality $\kappa$, there exists a colouring $c:G\rightarrow G$ such that
for every positive integer $n$, every $X_0,\ldots,X_n \in[G]^\kappa$, and every
$\gamma\in G$, there are $x_0\in X_0,\ldots, x_n\in X_n$ such that
$c(x_0+\cdots+x_n)=\gamma$. Then $\circledast_\kappa$ holds for unboundedly
many uncountable cardinals $\kappa$, and it is consistent that
$\circledast_\kappa$ holds for all regular uncountable cardinals $\kappa$.

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The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
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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: 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
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The categories of this paper are: econ
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The categories of this paper are: patt-sol