I'm considering two families, $F_1$ and $F_2$, of lines in the plane with $\vert F_1 \vert= N_1$ and $\vert F_2 \vert =N_2$. The families are such that if we pick $g \in F_1$ and $l \in F_2$ we get that the intersection of this two lines is a point. Further we know that at most $n_1$ of the lines in $F_1$ can intersect in a point, and $n_2$ for $F_2$. I'm looking for a lower bound of the number of intersection points.
A trivial lower bound would be 1 but we can not achive this if $n_i < N_i$, since then not all lines can go through the same point. Can we in this case establish a better lower bound dependent on $N_1, N_2, n_1$ and $n_2$?
I tried to state the problem in a more combinatorial way here: Minimum for combinatorial sortingproblem
But maybe the more geometric version is more appealing for you. Thanks in advance!