Assume κ is a strongly compact cardinal, T is a theory in a fragment F of L_κω over a language L, and κ' = max(κ,|F|). Assume T is categorical in λ. If λ is a successor cardinal and λ > ((κ') ^{< κ} )⁺, then T is categorical in every cardinal ≥ min(λ,BETH_{(2^κ' )⁺}).

This is one of the main results from the 1990 paper by Makkai and Shelah, "Categoricity of theories in L_κω, with κ a compact cardinal". This result should be considered a step towards settling Shelah's categoricity conjecture for L_ω1ω, which says the following.

Given a countable language L and T a theory in L_ω1ω, if T is categorical in λ>BETH_ω1, then T is categorical in every cardinal ≥BETH_ω1.