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 \([\mathbb {N}]^\mathbb {N}\) with finite (or, equivalently, \(\le 3\)) Borel chromatic number form a \(\varvec{\Sigma }^1_2\)-complete set. This answers a question of Kechris and Marks and strengthens several earlier results.

中文翻译：

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Borel图的复杂性问题

我们表明，对于具有无限Borel色数的Borel图，没有简单的（例如，有限的或可数的）基础。实际上，证明了（\ [[\ mathbb {N}] ^ \ mathbb {N} \）上移位图的闭合子图具有有限的Borel色数（或等效于\（\ le 3 \））形成\（\ varvec {\ Sigma} ^ 1_2 \）-完整集。这回答了Kechris和Marks的问题，并加强了一些较早的结果。