Total degree of intersection points of $n$ lines in the plane Here's a conjecture I have.  Can anyone prove or disprove it?
Given $n$ distinct straight lines in the plane, the total degree of any $n$ (or less) intersection points is O$(n)$ (where the degree of a point is the number of lines containing it).
Update:  I meant the question as Justin Smith understood it.  Pick any $n$ or less intersection points (actually they don't even have to be intersection points---just points in the plane).  Then for each of these points, count how many lines contain it.  Add up these numbers.  The conjecture was that the sum will always be O$(n)$. Note that $n$ is both the number of lines and an upper bound on the number of points you are allowed to select.
 A: The sum of "degrees" of a set of points is the total number of "incidences" for those points.  There're numerous related bounds, the most famous of which (and applicable to this question)  is Szemeredi-Trotter. http://en.wikipedia.org/wiki/Szemer%C3%A9di%E2%80%93Trotter_theorem
This bound applies even when you only select a subset of the lines' intersection points.
And that bound is tight, i.e., there are known arrangements of lines that attain the asymptotic bound.
Tao has a demonstration of points and lines that achieves Omega(n^(4/3)) incidences here: http://terrytao.wordpress.com/2009/06/12/the-szemeredi-trotter-theorem-and-the-cell-decomposition/
A: Unless I'm missing something, I don't think this holds: write numbers $1$ to $n$ on line $A$ in the natural order, then the same numbers but in the reverse order on a parallel line $B$, and connect $i$ on $A$ to $i$ on $B$ by a straight line, spacing numbers as required so that all intersections have degree $2$. Then you have ${n \choose 2}$ intersections of degree $2$.
