A power improvement in the Heilbronn triangle lower bound

There are absolute constants $\eta ,c_1>0$ such that, for every sufficiently large integer n, one can choose n points in the unit square so that every triangle they determine has area at least $c_1{n}^{-2+\eta }$. Thus the almost n${}^{-2}$ upper-bound formulation of Heilbronn's triangle problem is false. The exponent η is fixed but extremely small.