Joint work with Tanmay Inamdar.
Abstract. We find new sufficient conditions for the existence of entangled linear orders. The constructions use anti-Ramsey colourings for trees, and they provide a fine control on the extent of entangledness. As an application, from a strongly Luzin set we obtain for every positive integer $n$, an $(\aleph_1, n)$-entangled linear order that is not $(\aleph_1, n+1)$-entangled. The result is more general, and answers a question of Carroy, Levine, and Notaro. As another, we improve a theorem of Shafir and Shelah concerning entangled linear orders above the first singular strong limit cardinal.
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