For every n≥2, the statement 'there is an n-entangled set of reals but no (n+1)-entangled set' is consistent with PID plus Martin's Axiom.
On the consistency of some partition theorems for continuouscolorings,andthestructureof ℵ1-denserealordertypes
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There may be an $n$-entangled set but no $n+1$-entangled sets
For every n≥2, the statement 'there is an n-entangled set of reals but no (n+1)-entangled set' is consistent with PID plus Martin's Axiom.