Tight lower bounds for the size of epsilon-nets
According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an epsilon-net of size O (1/epsilon log 1/epsilon). Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VC-dimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest epsilon-nets is superlinear in 1/epsilon, were found by Alon (2010). In his examples, every epsilon-net is of size Omega (1/epsilon g(1/epsilon)), where g is an extremely slowly growing function, related to the inverse Ackermann function.