Consequences of lower bounds for $\epsilon$-nets on approximation
If a range space has $\epsilon$-net of size $f(1/\epsilon)$, then the integrality gap of the fractional hitting set (or set cover) is $f(1/\epsilon)/(1/\epsilon)$. See the work by Philip Long (here [The Even etal. work is later than this work, and rediscover some of his stuff]). See also the slides 13-16 here. 

In short, having non-linear $\epsilon$-nets, indicates that approximating the relevant hitting-set/set cover problem within  better than a constant factor is going to be very challenging.