Erdős graphs resolve fine's canonicity problem

Bulletin of Symbolic Logic 10 (2):186-208 (2004)
We show that there exist 2 ℵ 0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of Thomason). The constructions use the result of Erd $\H{o}$ s that there are finite graphs with arbitrarily large chromatic number and girth
Keywords Boolean algebras with operators   modal logic   random graphs   canonical extension   elementary class   variety
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DOI 10.2178/bsl/1082986262
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Marcus Kracht (2011). Technical Modal Logic. Philosophy Compass 6 (5):350-359.

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