They are named after Mikhail Yakovlevich Suslin.
The diamond principle, a consequence of V=L, implies that there is a Suslin tree, and Martin's axiom MA(ℵ1) implies that there are no Suslin trees.
Jensen (1972) showed that if V=L then there is a κ-Suslin tree for every infinite successor cardinal κ.
Whether the Generalized Continuum Hypothesis implies the existence of an ℵ2-Suslin tree, is a longstanding open problem.
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