### On the number of $L_{\infty,\omega_1}$-equivalent non-isomorphic models

by Shelah and Vaisanen. [ShVs:646]

Transactions American Math Soc, 2001

We prove that if ZF is consistent then ZFC + GCH is
consistent with the following statement: There is for every k <
omega a model of cardinality aleph_1 which is
L_{infty, omega_1}-equivalent to exactly k non-isomorphic
models of cardinality aleph_1 . In order to get this result we
introduce ladder systems and colourings different from
the ``standard'' counterparts, and prove the following purely
combinatorial result: For each prime number p and positive
integer m it is consistent with ZFC + GCH that there is a
``good'' ladder system having exactly p^m pairwise
nonequivalent colourings.

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