Before revision:  With a slight modification of Elekes' construction, we show that it can give a better lower bound of C _{SzTr} \ge 1, i.e., I(n,l) \ge (nl)^{2/3 . Furthermore, we analyze a different construction given by  3 } , and show its constant of proportionality to be even better, C _{\mathrm{SzTr}} \ge 3/(2^{1/3}\pi^{2/3}) = 1.11.
After revision:  With a slight modification of Elekes' construction, we show that it can give a better lower bound of C _{SzTr} \ge 1, i.e., I(n,l) \ge (nl)^{2/3 . Furthermore, we analyze a different construction given by  Erd } , and show its constant of proportionality to be even better, C _{\mathrm{SzTr}} \ge 3/(2^{1/3}\pi^{2/3}) = 1.11.