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Recall that a group $G$ admits a $(m+n)$-paradoxical decomposition if there exist positive integers $m$ and $n$, a partition $\{P_1,\dotsc,P_m,Q_1,\dotsc,Q_n\}$ of $G$ and elements $x_i, y_j$ of $G$ such that $$ G=\bigcup_{i=1}^m x_iP_i=\bigcup_{j=1}^ny_jQ_i$$ The minimal possible value of $m + n$ in a paradoxical decomposition of $G$ is the Tarski number of $G$.

Let the Tarski number of a group be 7. Does this group admit (2+5)-paradoxical decomposition, (3+4)-paradoxical decomposition or both?

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    $\begingroup$ There are obviously infinitely many questions of this sort one could ask, although many of them are surely naïve. Why are the cases $7 = 2 + 5 = 3 + 4$ particularly interesting? $\endgroup$
    – LSpice
    Commented Aug 23, 2020 at 15:05
  • $\begingroup$ @LSpice That is because I have some ideas to construct a group which have a $(m+n)$-paradoxical decomposition. $\endgroup$
    – MSMalekan
    Commented Aug 23, 2020 at 17:51
  • $\begingroup$ @LSpice Also, the question of the existence of a group with tarski number 7 is open yet. $\endgroup$
    – MSMalekan
    Commented Aug 23, 2020 at 18:00

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