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We use Ramsey's theorem to generalise a result of L. Babai and
T.S. Sos on Sidon subsets and then use this to prove that for an
integer $n>1$ the class of groups in which every infinite subset
contains a rewritable $n$-subset coincides with the class of
groups in which every infinite subset contains n mutually disjoint
non-empty $X_1,\ldots, X_n$ such that $X_1\ldots X_n\cap X_{\sigma
(1)} \ldots X_{\sigma(n)} \neq \emptyset$ for some non-identity
permutation $\sigma$ on the set $\{1,\ldots, n\}$.
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