incidence combinatorics

the method from algebraic geometry and algebraic topology have a effect on incidence combinatorics this years.espatialy on finite field case.there is some examples of the achievement follow this idea.

Dvir-Finite Kakeya conjecture

Guth-Katz-Erdos Distance problem

there is a example with classical algebraic geometry,cubic curve in fact.

Ben Green:

$ P\subset R^2$,$ P$ is a set consist with n points.

A k-rich line is a line in $ R^2$ which  contain k points of $ P$

$ N_k=$#k-rich lines,$ k\geq 2$.we call 2-rich line as original line.

there is a classical theorem:

Sylvester-Gallai theorem:if the points in $ P$ is not collinear,then $ N_2\geq 2$.

this theorem is not true in other fields.

there is a lots of counterexample.

the original proof of sylvester-Galli theorem:

find the pair of point and line minimize the distance from the point to the line.if the line is not original we can get a contradiction!

this proof is very pretty,but too clever to extend to a system method to deal with similar problem in incidence geometry…