Let I(n,l) denote the maximum possible number of incidences between n points and l lines. It is well known that I(n,l) = \Theta( (nl)^{2/3 + n + l) 2,3,7 . Let C _{SzTr} denote the  constant of proportionality of the (nl)^{2/3 term. The known lower bound, due to Elekes 2 , is C _{SzTr} \ge 2^{-2/3} = 0.63. 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 %DIFDELCMD < \erdos [%%% 3 } , and show its constant of proportionality to be even better, C _{\mathrm{SzTr}} \ge 3/(2^{1/3}\pi^{2/3}) = 1.11.
Let I(n,l) denote the maximum possible number of incidences between n points and l lines. It is well known that I(n,l) = \Theta( n^{2/3 + n + l) . Let c _{SzTr} denote the lower bound on the constant of proportionality of the n^{2/3 term. The known lower bound, due to Elekes , is c _{SzTr} \ge 2^{-2/3} = 0.63. 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 n^{2/3 . Furthermore, we analyze a different construction given by %DIFDELCMD < \erdos [%%% Erd \H o}s , and show its constant of proportionality to be even better, c _{\mathrm{SzTr}} \ge 3/(2^{1/3}\pi^{2/3}) \approx 1.11.
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