1 Classic DG.dvi

We use standard asymptotic notation. That is, f(n) = O(g(n)) implies that there exist constants c, n0, such that for any n ≥ n0, we have f(n) ≤ c · g(n). For example, 10n + 1000 = O(n) holds since we can take c = 100 and n0 = 20. Similarly, f(n) = Ω(g(n)) implies that there exist constants c, n0, such that for any n ≥ n0, we have f(n) ≥ c · g(n). Finally, f(n) = Θ(g(n)) implies that both f(n) = O(g(n)) and f(n) = Ω(g(n)) hold. Given a set P of points and a set L of lines, both in R, an incidence is a pair (p, l) ∈ P ×L such that the point p is contained in the line l. We denote by I(P,L) the number of incidences in P × L. For any m,n, Erdős constructed a set P of m points and a set L of n lines with Θ(mn + m + n) incidences. Erdős also conjectured that no point-line configuration has an asymptotically larger number of incidences. This conjecture has been proven by Szemerédi and Trotter [15] in 1983.