2 e4 omega gambit doubt its sound
1.d4 Nf6 Gambiteers...
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Lots of aggressive and gambity lines in the Tromp, depending on which lines black enters.
You can play 2.Nc3 and after 2...d5 play 3.e4, hoping for a Blackmar Diemer Gambit after 3...dxe4 4.f3
As you've probably figured out, Black can play 3...Nxe4, the Hübsch Gambit, which can be annoying.
So if you want to play 2.Nc3, have something ready for the Hübsch Gambit. One recommendation is 4.Nxe4 dxe4 5.Bf4!?, a relatively rare move compared to the main line with 5.Bc4.
Another option to steer the game into a BGD is with 2.f3, hoping for 2...d5 3.e4 dxe4 with a transposition.
Unfortunately, it's not so simple because after 2.f3 Black has a few option you'll need to be ready for:
a) 2...c5. Now 3.d5 is an option with a Benoni structure, but we don't have a gambit. If anything, Black often ends up going for a Benko-like gambit with a timely b5. Another option is just to continue your development, say 3.Nc3 and let Black play cxd4. Of course, that has the drawback of being forced to recapture with the Queen, but maybe after Qxd4 Nc6 you could put your Queen on h4, followed by Bg5, and 0-0-0.
b) 2...g6. Again, we can't really gambit anything at this point. We'll probably play 3.e4, and after 3...Bg7 we'll need to decide between continuing our development with 4.Nc3 or 4.c4. The drawback here is that after 4...d6, we're back in either the Pirc or the KID, so you'll need to brush up on the f3 line vs the Pirc if you want to play 4.Nc3 (usually a with Be3+Qd2 setup) and the Saemisch vs the KID if you prefer 4.c4.
c) 2...d5 (or e6) 3.e4 e6 (or d5). Does this look familiar? It's basically a sort of French Defence now. You can continue in typical French style with 4.e5 Nfd7 5.f4 but while it's probably fine, it's probably not the type of position you wanted to play. Another option is 4.Nc3 but after 4...Bb4 or c5 I don't see a good gambit line. Lastly 4.Be3 is an option, hoping for 4...dxe4 5.Nd2, the Alapin-Diemer Gambit.
What are some options for gambit lines FOR WHITE after d4 Nf6? I know black has options such as the Budapest gambit but not much comes to mind for white. It'd be nice to play with some reckless abandon after such an opening.
Obviously if white is out for blood he can play g4 at anytime, but I'm hoping for something with more promising chances and has a bit more study behind it.
EDIT: The gambit doesn't have to be on move 2. It could be move 7 after a normal line of the KID.