A forcing construction yields a chain of length ω₃ in [ω₁]^{ω₁} that is increasing modulo finite sets.
Forcing with symmetric systems of models of two types.arXiv preprint arXiv:2210.12741v2 [math.LO]
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Long Strong Chains of Subsets of $\omega_1$
A forcing construction yields a chain of length ω₃ in [ω₁]^{ω₁} that is increasing modulo finite sets.