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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
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.