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Simple proofs of classical theorems in discrete geometry via the GuthKatz polynomial partitioning technique
 DISCRETE COMPUT. GEOM
"... Recently Guth and Katz [16] invented, as a step in their nearly complete solution of Erdős’s distinct distances problem, a new method for partitioning finite point sets in R d, based on the Stone–Tukey polynomial hamsandwich theorem. We apply this method to obtain new and simple proofs of two well ..."
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Recently Guth and Katz [16] invented, as a step in their nearly complete solution of Erdős’s distinct distances problem, a new method for partitioning finite point sets in R d, based on the Stone–Tukey polynomial hamsandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemerédi–Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at selfcontained, expository treatment. We also mention some generalizations and extensions, such as the Pach–Sharir bound on the number of incidences with algebraic curves of bounded degree.
The discrepancy method in computational geometry
 In Handbook of Discrete and Computational Geometry
, 2004
"... Discrepancy theory investigates how uniform nonrandom structures can be. For example, given n points in the plane, how should we color them red and blue so as to minimize the difference between the number of red points and the number of blue ones within any disk? Or, how should we place n points in ..."
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Cited by 5 (0 self)
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Discrepancy theory investigates how uniform nonrandom structures can be. For example, given n points in the plane, how should we color them red and blue so as to minimize the difference between the number of red points and the number of blue ones within any disk? Or, how should we place n points in the unit square