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IMO, I would have no problem framing the floor with 2x4, but I think the choice of using 2x4s or 2x6s as the vertical post is not a good one.
The reason is that they are plenty strong vertically, but at only 1.5" wide in the "2x" direction, provide pretty mediocre bending resistance from side-to-side. The directions you link to use a bunch of corner brackets/ties to try to tackle this. They have also used additional 2x4s on the perpendicular direction to help prevent side to side bending.
The cost for the 2 8' 4x4 posts you'd need (get them cut in half at Home Depot) is peanuts and will make for a much sturdier/less bouncy feeling platform. It will probably cost less than the extra brackets to beef up the 2x6 corners.
-Mark
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I'm old now - boring street cars and sweet race cars.
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