Finite colorings of G^2 for countable amenable G contain monochromatic right corners with IP* return times, and FC-groups satisfy the analogous statement in every dimension.
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Van der Waerden type theorem for amenable groups and FC-groups
Finite colorings of G^2 for countable amenable G contain monochromatic right corners with IP* return times, and FC-groups satisfy the analogous statement in every dimension.