Matchings vs hitting sets among half-spaces in low dimensional euclidean spaces

Let F be any collection of linearly separable sets of a set P of n points either in R, or in R . We show that for every natural number k either one can find k pairwise disjoint sets in F , or there are O(k) points in P that together hit all sets in F . The proof is based on showing a similar result for families F of sets separable by pseudo-discs in R. We complement these statements by showing that analogous result fails to hold for collections of linearly separable sets in R and higher dimensional euclidean spaces. Department of Computer Science, Technion—Israel Institute of Technology, Haifa 32000, Israel, and Max Planck Institute for Informatics, Saarbrücken, Germany. [email protected]. Mathematics Department, Technion—Israel Institute of Technology, Haifa 32000, Israel. [email protected]. Supported by ISF grant (grant No. 1357/12).