Main Gap Phenomenon

Shelah's Main Gap Theorem states that for all first-order, complete theories, T, in a countable language, we have that either $$I(T,\aleph_\alpha)=2^{\aleph_\alpha}$$ or $$I(T,\aleph_\alpha)<\beth_{\omega_1}(\alpha)$$ This result, while beautiful, leaves some questions. Can analogues of this result be extended to larger languages? Do other theorems exists for logics like $L_{\omega_1 \omega}$ or $L_{\omega_1 \omega}(Q)$? Do there exist any logics without the 'gap phenomenon'?

Any resources would be helpful. Thanks!

• It seems that the problem for $L_{\omega_1,\omega}$ is still open, as well as even more modest problems (see the discussion on the end of page 43 here): shelah.logic.at/files/702.pdf – Haim Jul 13 '14 at 3:27
• @Haim: Ahh, your right. On page 25, he remarks that it is still a problem to prove the main gap for uncountable theories! – Kyle Gannon Jul 13 '14 at 4:04

For full $L_{\omega_1,\omega}$ this seems to be super hard as it may require proving Shelah's categoricity conjecture for $L_{\omega_1,\omega}$ which despite of many hundreds of pages of approximation is still open.