... RSC[k, r, d] of cardinality n to one of cardinality n + or greater Theorem 8.1 If for some r ≥ k ≥ 3, and some d, we have a RSC[k, r, d] of cardinality n, and for some r , k − ≤ r ≤ r − 2, and d , ... cubic acute dr -set of cardinality nr , where n1 is the least of the nr , and if, for some dimension dZ , we have a cubic acute dZ -set of cardinality nZ , where m nZ ≥ nr , r=2 then a cubic acute ... result of Erd˝s and F¨ redi by a factor of ≈ 1.089 o u This is achieved by a slight improvement in the choice of parameters This proof can also be found in [AZ3] Proof: Let m = √ √ d and randomly...