Open Journal of Discrete Mathematics
Volume 3, Issue 3 (July 2013)
ISSN Print: 2161-7635 ISSN Online: 2161-7643
Google-based Impact Factor: 0.64 Citations
On Some Numbers Related to the Erdös-Szekeres Theorem ()
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ABSTRACT
A crossing family of segments is a collection of segments each pair of which crosses. Given positive integers j and k,a(j,k) grid is the union of two pairwise-disjoint collections of segments (with j and k members, respectively) such that each segment in the first collection crosses all members of the other. Let c(k) be the least integer such that any planar set of c(k) points in general position generates a crossing family of k segments. Also let #(j,k) be the least integer such that any planar set of #(j,k) points in general position generates a (j,k)-grid. We establish here the facts 9≤c(3)≤16 and #(1,2)=8.
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