A complexity problem for Borel graphs

Creators: Todorčević, Stevo; Vidnyánszky, Zoltán

Abstract

We show that there is no simple (e.g. finite or countable) basis for Borel graphs with infinite Borel chromatic number. In fact, it is proved that the closed subgraphs of the shift graph on [N]^N with finite (or, equivalently, ≤3) Borel chromatic number form a Σ1/2-complete set. This answers a question of Kechris and Marks and strengthens several earlier results.

Additional Information

© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Received 04 March 2019; Accepted 25 March 2021; Published 10 May 2021. We would like to thank Benjamin Miller for the inspiring discussions, questions and for pointing out a way to prove the analytic-hardness of non-dominating sets using only classical tools (see Lemma 4.6). We are also very grateful to Slawomir Solecki, Alexander Kechris, Márton Elekes and Jan Grebík for their help, valuable comments and suggestions. Open access funding provided by University of Vienna. The first author's research was partially supported by grants of NSERC (455916) and CNRS (IMJ-PRG UMR7586). The second author was partially supported by the National Research, Development and Innovation Office – NKFIH, Grants no. 113047, no. 129211 and FWF Grants P29999 and M2779.

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