Thm [Alon]: For any C and eps there is a set of n points in the plane such that any weak eps-net for lines has >C/eps points.
Proof: First we prove that there is no (strong) eps-net.
Use Density HJ for k>2C and p=1/2, while n=k^d and eps=k/k^d, and project it to a generic plane.
Proof for weak eps-nets will follow from this, but first we need:
Lemma: Combinatorial lines only intersect at points of [k]^d. (Proof not hard.)
Cor: After projection to generic plane, at any point at most two combinatorial lines cross.
So if we have a weak eps-net, we can get a strong one that is at most twice as big.
Hadwiger-Debrunner (p,q)-problem: Given finite family of convex sets in R^d such that from any p at least q>=d+1 intersect.
Is it true that we can pierce them all with C(p,q) points?
Note that if q=d, then generic hyperplanes give counterexample.
Thm [Alon-Kleitman]: For q>=d+1 C(p,q) points are enough.
Note that C(p,q)>=p-q+1 and if p(d-1)<(q-1)d, then we have equality (Ex to prove).
Proof: It is enough to prove the statement for q=d+1.
Using Weak eps-net thm's cor, it is enough to prove that tau*=nu* is bounded.
To bound nu*, take a psi:F->[0,1] fractional packing for which sum_F psi(F)>nu*/2 and psi(F) is rational for every F.
Take a large D for which psi(F)=m(F)/D for every F, where D and m(F) are integers.
Define the multifamily MF by taking m(F) copies of each F, so |MF|>=Dnu*/2.
For MF we have that from any (p-1)d+1 sets there are d+1 that intersect.
Using double counting, at least alpha of the d+1-tuples intersect, so using fractional Helly, at least beta|MF| sets intersect,
but this implies that beta|MF|/D original weighted sets intersect, so 1>=beta|MF|/D=betanu*/2, we are done.
Bit about k-levels (for k=1 parabola, Davenport-Schinzel), then k-sets, halving lines.