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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.
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.