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!



Main Gap Phenomenon
Extending Shelah's main gap to non first-order or even to first-order theories in an uncountable language is a major hard open problem. 


I am familar with couple of extensions to non f.o. (both papers are available from my web page):


1. Rami Grossberg and Bradd Hart. The classification theory of excellent classes, Journal of Symbolic Logic, 54, (1989) 1359--1381.
2. Rami Grossberg and Olivier Lessmann. Abstract decomposition theorem and applications, Contemporary Mathematics, Vol 380, (2005), AMS, pp. 73--108.


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.


