Thm: Exists 1/r-net of size Cdr log r, if VC-dim(F)<=d, where C<20 and C->1 as r-> infty.
For proof of thm, we need following def: pi(m):= max number of subsets cut out by F from a set of size m.
Lemma: pi(m)<= sum_{i=0}^d {m\choose i}<(em/d)^d.
Proof: Downshift.
Proof of thm: Take s=Cdr log r and select s points at random: N and another s points at random: M.
(If measure is not uniform, we choose according to that probability measure!)
E0: Event that N is not a 1/r-net and E1: Event that for an S from F: |S\cap N|=0 but |S\cap M|>=s/2r.
We have Pr[E0]>=Pr[E1] (supposing that all S have size at least 1/r). We prove that also 2Pr[E1]>=Pr[E0].
Fix N. If it is a 1/r-net, both events are surely 0. If it is not, fix an S that avoids it.
Now select M at random - each point of M has a 1/r chance of hitting S and rest is calculation.
So now it is enough to bound Pr[E1]. We select first 2s points, then divide it to N and M.
For any 2s points, sets of F intersect them in at most (e2s/d)^d different subsets (using lemma).
For each subset S, the chance that |S\cap N|=0 but |S\cap M|>=s/2r is at most (1/2)^{s/2r}=r^{-Cd/2}.
Using union bound and taking 1/d power, we get (e2Crlog r)/r^{C/2}<1 if C is big.
Last time we proved Cor: tau(F)<=Cdtau*(F) log(tau*(F)).