As is usually the answer with business, it's the Commerce Clause. ("Congress shall have the power... to regulate commerce... among the several states")

It's a basically open-ended constitutional grant to regulate business, with the only caveat being that it's restricted to activities which affect the flow of commerce across state lines.

That would give Congress the power to regulate these contracts; it wouldn't make them unconstitutional automatically, unless there's some argument I'm missing.

True enough, I'm not sure why the word 'unconstitutional' was used. This seems to be a pretty clear case of "laws enabled by the commerce clause", so all it would take is a repeal of Sherman to change the situation.