Abstract
Call a set \({A \subseteq \mathbb {R}}\) paradoxical if there are disjoint \({A_0, A_1 \subseteq A}\) such that both \({A_0}\) and \({A_1}\) are equidecomposable with \({A}\) via countabbly many translations. \({X \subseteq \mathbb {R}}\) is hereditarily nonparadoxical if no uncountable subset of \({X}\) is paradoxical. Penconek raised the question if every hereditarily nonparadoxical set \({X \subseteq \mathbb {R}}\) is the union of countably many sets, each omitting nontrivial solutions of \({x - y = z - t}\). Nowik showed that the answer is ‘yes’, as long as \({|X| \leq \aleph_\omega}\). Here we show that consistently there exists a counterexample of cardinality \({\aleph_{\omega+1}}\) and it is also consistent that the continuum is arbitrarily large and Penconek’s statement holds for any \({X}\).
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This paper is dedicated to Rich Laver, in memory of his mathematical power and integrity, and for being so polite about it.
Research supported by the Hungarian National Research Grant OTKA K 81121.
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Komjáth, P. A remark on hereditarily nonparadoxical sets. Arch. Math. Logic 55, 165–175 (2016). https://doi.org/10.1007/s00153-015-0463-6
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DOI: https://doi.org/10.1007/s00153-015-0463-6