Joint work with Tanmay Inamdar.
Abstract. We find new sufficient conditions for the existence of entangled linear orders, among others, improving a theorem of Shelah and Shafir. The constructions go through an anti-Ramsey colouring for trees and they provide a fine control on the extent of entangledment. 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 it answers a question of Carroy, Levine and Notaro.
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