20 dead in Orlando gay club shooting

[GUBU]

As an attempt to try and get both sides to agree on something, can we at least agree that there is a problem with gun crime in this country which doesn't happen in other countries?

Plenty of Scandinavian countries have high gun ownership, but low gun crime. This suggests to me that some balance can be made where guns aren't banned outright, but gun crime is reduced.

So, what solutions would people suggest that don't explicitly violate the 2nd amendment?

A big part of a proper gun control system is licensing the owners. That is what happens in Europe. To use your Sweden example, this is what you need to do to be able to obtain a license:

1) Show a genuine requirement for ownership
2) Be over eighteen, with a few exceptions
3) Pass a background check which goes into criminal and mental issues
4) Have no history of domestic violence (if it happens after you get your license you're legally required to surrender your guns)
5) Have attended and passed the required firearms safety courses.

More information is from here. Please note, that like most of the rest of Europe Sweden doesn't have an automatic right to own guns (frankly, if the constitution were read right neither would US citizens. The second amendment was a provision to set up a militia rather than go to the expense of either a mercenary or standing army).