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A superlinear lower bound on the number of 5-holes

Publikace na Matematicko-fyzikální fakulta |
2017

Tento text není v aktuálním jazyce dostupný. Zobrazuje se verze "en".Abstrakt

Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P.

For a positive integer n, let h_5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. We show that h_5(n) = Omega(n log^{4/5} n), obtaining the first superlinear lower bound on h_5(n).

The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line l into two subsets, each of size at least 5 and not in convex position, then l intersects the convex hull of some 5-hole in P.

The proof of this result is computer-assisted.