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Yep, not thinking specifically 2x4 / 4x4 etc just using those numbers for illustration of the question 'what is strongest'.
In my early days of thinking about it head I imagine a kind of L / J shape for the slat panel to rest on. So a 4x4 (or whatever) with a third cut out of one corner so there's still a solid structure but there's not so much frame on show. Or something.
CYOA-
It would be a huge waste of money to buy a beam that big and cut so much away. Buy a nice thick board of your preferred wood, and glue + screw a cheap structural timber to the back of it to add stiffness and rest the slats on.
In general, more is stronger, but more thickness in the vertical direction is more important (to the power of 3).
chiroshi
A hardwood 2x4 will be on the limit for withstanding bed time activities. A 2x6 would be better than a 4x4 and I'm sure still cheaper. Consider that you'll need to add rails for the slats which will add a bit of stiffness, and if you have a really wide rail, it will be quite a step between the frame and the mattress.
What will matter most is the joinery. Wobbly joints are more common than bendy frames.