Now we definitely can improve things by using "improved random" constructions. Here is a simple example.
Randomly fill in an nXm matrix with {-1,0,1}. Let C be the number of triples of columns which sum (mod 3) to 0, not all three columns the same. Remove one column from each such triple from the matrix, result being an nX(m-C) trit-matrix (or the horizontal size could be larger than m-C if two bad triples overlap) which is a SET. Again the expected number of triples of columns which sum (mod 3) to 0, not all three columns the same, is (m^2+3*m-4)*(m/6)/3^n. There must exist a random matrix fill which causes C to be its expected value or less, hence F(n)>=m-C>=m-(m^2+3*m-4)*(m/6)/3^n. (In fact ">".) This yields THEOREM F(n) > sqrt((216*T^3+756*T^2+882*T+343)/T^2)/(9*sqrt(3))-(T+1)/T where T=3^n. This also grows exponentially, but now with the improved growth constant 3^(2/3) =approx= 2.080.
--correction. The above formula, obtained by maximizing m-(m^2+3*m-4)*(m/6)/T by choice of m>=1, actually grows ultimately exponentially with growth constant 3^(1/2) =approx= 1.732. Not 2.080.