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In [2] we consider the statement that if the Continuum Hypothesis holds then the plane can be colored with countably many colors such that no right angled triangle is monocolored. The proof given there is incomplete: at a point the authors describe some conditions and state that “These requirements can be met by an inductive selection of colors.” Unfortunately, this is not true. With an essential modification of the argument, with Balázs Bursics, we are going to publish a full argument (see [1]).
References
Bursics, B., Komjáth, P.: A coloring of the plane without monochromatic right triangles (2023). arXiv:2302.12215
Erdős, P., Komjáth, P.: Countable decompositions of \(R^2\) and \(R^3\). Discrete Comput. Geom. 5(4), 325–331 (1990)
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Komjáth, P. Corrigendum to “Countable Decompositions of \(R^2\) and \(R^3\)”. Discrete Comput Geom 71, 1165 (2024). https://doi.org/10.1007/s00454-023-00542-9
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DOI: https://doi.org/10.1007/s00454-023-00542-9