All Questions

Suppose a group $G$ acts on an infinite set $X$ and $X$ has no non-empty $G$-paradoxical subsets. Is it possible for $X$ to have non-trivial $G$-paradoxical subsets modulo finite sets? I.e., can there ...

The Banach-Tarski paradox states that for a solid ball in 3‑dimensional space, there exists a decomposition into a finite number of disjoint subsets, which can then be put back together in a different ...

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 $G$ ...

Two questions, the first: What is the smallest non negative integer that we do not know yet is the Tarski number of a group? The second question is the same as in the title: What is the latest ...