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  "body": "\n## Statement\n\nThere is an absolute constant $C \u003e 0$ with the following property.\nLet $P$ be a finite set of points and $L$ a finite set of lines in the [[Euclidean plane|0XW7V9ZZ05]] $\\R^2$.\nLet $I(P, L)$ be the number of pairs $(p, \\ell) \\in P \\times L$ with $p \\in \\ell$.\nThen\n$$I(P, L) \\le C \\left( \\abs{P}^{2/3} \\abs{L}^{2/3} + \\abs{P} + \\abs{L} \\right).$$\n\n## Remarks\n\nSzékely gave a short proof with the crossing number inequality.\nDraw a graph whose vertices are the points and whose edges join consecutive points on each line.\nIt has at least $I - \\abs{L}$ edges and at most $\\abs{L}^2$ crossings, and the crossing number inequality gives the bound.\nThe bound is sharp up to the constant.\nTake the grid $\\set{1, \\dots, k} \\times \\set{1, \\dots, 2k^2}$ and the lines $y = ax + b$ with $1 \\le a \\le k$ and $1 \\le b \\le k^2$.\nEach such line meets the grid in $k$ points, so there are $k^4$ incidences, with $2k^3$ points and $k^3$ lines.\nThe theorem fails over finite fields.\nIn $\\F_q^2$, the $q^2$ points and the $q^2 + q$ lines have $q^3 + q^2$ incidences, which is more than a constant times $q^{8/3}$ for large $q$.\n\n## Explanations\n\n### L1 human\n\nDraw some points and some straight lines in the plane, and count how many times a point lies on a line.\nThe theorem gives the largest possible count, up to a constant factor.\nWith $n$ points and $n$ lines, there are at most about $n^{4/3}$ such incidences, far fewer than the trivial $n^2$.\n\n### L2 human\n\nThe Szemerédi–Trotter theorem is the basic result of incidence geometry.\nA corollary: $n$ points determine at most $O(n^2/k^3 + n/k)$ lines that contain at least $k$ of the points.\nIt has applications to the [[Erdős distinct distances problem|4H31EM5T0C]] and to sum-product estimates in [[additive combinatorics|3NM362A2JG]].\nIts proof uses the topology of the real plane, which explains why it fails over finite fields.\n",
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