A question about the number of intersections of lines in $R^{3}$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:30:03Z http://mathoverflow.net/feeds/question/61153 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61153/a-question-about-the-number-of-intersections-of-lines-in-r3 A question about the number of intersections of lines in $R^{3}$ rrrq 2011-04-09T17:25:29Z 2011-04-10T02:17:41Z <p>Suppose I have n lines in $R^{3}$ with the conditions that: no 3 lines in one plane, no 3 lines intersect at one point, for fixed 2 lines, no 3 lines intersect these 2 lines at the same time. what is the up bound of the number of intersections? The up bound $n^{\frac{3}{2}}$ is a simple corollary of Guth-Katz's paper or one can prove it directly by algebraic method. Is it possible to establish the up bound like $n^{\frac{4}{3}}$ or some better one?</p> <p>The up bound will also be a up bound for Erdos's unit distance problem in $R^{2}$.</p> http://mathoverflow.net/questions/61153/a-question-about-the-number-of-intersections-of-lines-in-r3/61178#61178 Answer by JSE for A question about the number of intersections of lines in $R^{3}$ JSE 2011-04-10T02:17:41Z 2011-04-10T02:17:41Z <p><a href="http://terrytao.wordpress.com/2010/11/20/the-guth-katz-bound-on-the-erdos-distance-problem/" rel="nofollow">Terry's discussion of the Gutz-Katz paper on his blog</a> gives an example showing that the Guth-Katz bound on incidences between lines in R^3 is sharp. (Look right after the statement of Theorem 5.)</p> <p>Your conditions are stronger than the ones there (e.g. you demand no more than 3 lines in a plane, where they only ask that no more than sqrt(N) lie in a plane) and I didn't check whether this example satisfy your conditions too. But that example is surely a good place to start.</p>