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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?

The up bound will also be a up bound for Erdos's unit distance problem in $R^{2}$.

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Maybe you could give a summary of or reference for the Guth-Katz paper and for the Erdos unit distance problem? – Gerry Myerson Apr 10 '11 at 0:14
The best summary of Guth-Katz paper I can think is the link in JSE's answer below, for unit distance problem, one can find reference in the reference of – user13289 Apr 10 '11 at 4:14
If the lines are indexed by $1,\ldots,n$ and the set of lines intersecting line $i$ is called $A_i$, an upper bound is the maximum of [\sum_{i=1}^n|A_i|,] subject to $|A_i\cap A_j|\leqslant 2$ for all $i\neq j$. But it might be obvious that this bound is worse than $n^{3/2}$. – Thomas Kalinowski Apr 10 '11 at 7:51
George Purdy (U. Cincinnati) is an expert on this general topic. He is giving a seminar @NYU tomorrow on this. Maybe contact him? – Joseph O'Rourke Apr 11 '11 at 13:23

Terry's discussion of the Gutz-Katz paper on his blog 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.)

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.

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That example can not satisfy the constions in my question, in fact "for fixed 2 lines, no 3 lines intersect these 2 lines at the same time" is the most important condition in the question, it is not easy for me to think some example with many intersections but satify that condition.... – user13289 Apr 10 '11 at 3:46
I thought of your condition as saying something like "no three lines in a quadric" -- is that not right? What about n lines in a (singly) ruled surface of degree n^{1/2}? – JSE Apr 10 '11 at 14:38
It looks like "no five lines in a quadric" but not exactly same. n lines in a (singly) ruled surface of degree $n^{\frac{1}{2}}$ is a situation appeared if one try to prove the up bound $n^{\frac{3}{2}}$, but still the full strength of that condition will not be used... – user13289 Apr 10 '11 at 15:26

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