New Existence Proofs for 2-Nets
نویسندگان
چکیده
We describe a new technique for proving the existence of small 2-nets for hypergraphs satisfying certain simple conditions. The technique is particularly useful for proving o( 1 2 log 1 2 ) upper bounds which is not possible using the standard VC dimension theory. We apply the technique to several geometric hypergraphs and obtain simple proofs for the existence of O( 1 2 ) size 2-nets for them. This includes the geometric hypergraph in which the vertex set is a set of points in the plane and the hyperedges are defined by a set of pseudo-disks. This result was not known previously. We also get a very short proof for the existence of O( 1 2 ) size 2-nets for halfspaces in R .
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