The Constant of Proportionality in Lower Bound Constructions of Point-Line Incidences

05/31/2017

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by   Roel Apfelbaum, et al.

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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) = Θ(n^2/3l^2/3 + n + l). Let c_SzTr denote the lower bound on the constant of proportionality of the n^2/3l^2/3 term. The known lower bound, due to Elekes, is c_SzTr> 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> 1, i.e., I(n,l) > n^2/3l^2/3. Furthermore, we analyze a different construction given by Erdős, and show its constant of proportionality to be even better, c_SzTr> 3/(2^1/3π^2/3) ≈ 1.11.