How can I support a desk without legs or brackets
This is a pretty standard construction for floating shelves/counter top.


 mount a cleat to the walls (the third wall is omitted for clarity sake). In this example it's a 1x4 but that will depend on how much weight it needs to bear.
 Create a torsion box leaving a socket around the edges to accept the cleat.
 mount the unit and apply a backsplash to cover the gap where the back and sides meet the wall. It's not impossible to scribe in a three sided surface but it's a tremendous pain in the neck.


I would recommend MDO over MDF as it has the strength to weight ratio of plywood but the surface of mdf. As for the thickness w.r.t the strength of the front edge... this is really pushing the limit of what you can support without a leg or integral steal beam and not experience serious deflection or failure.