The original version of the so-called "joints problem" consists of the following:
Let $L$ be a set of lines in $\mathbb{R}^{3}$. Determine the maximum number of "joints" determined by these lines, where by joint we understand a point of $\mathbb{R}^{3}$ lying on three lines from the set $L$ but which do not all lie in the same plane (i.e. they are non-coplanar).
The conjectured answer by Sharir was that this number of joints is $\leq C |L|^{\frac{3}{2}}$, for some positive constant $C$; this was proven by Guth and Katz using a rather simple polynomial mathod which easily generalies to $\mathbb{R}^{d}$ in which case the upper bound becomes $C |L|^{\frac{d}{d-1}}$. (references can be found very easily on google; for example, see http://www.dagstuhl.de/Materials/Files/09/09111/09111.SharirMicha.Other.pdf)
Now, it seems to me that this polynomial method does not generalize to finite fields; so, my question is, can we get some upper bound in this case? My thoughts for now are to use the graph $G$ having as vertices the lines and to connected them if the lines intersect. Then the number of joints is the number of triangles of $G$ minus $\frac{1}{2}\left(\binom{|L|}{2} - k\right)$, where $k$ is the number of distinct planes determined by the $|L|$ lines... but I feel that I'll be getting really weak bounds if I majorize this (using graph theoretic stuff about the number of triangles and Beck's theorem or related things for $k$).
So, any other ideas or knowledge about the finite field case in literature? Thanks.
