On the minimum size of a point set containing a 5-hole and double disjoint 3-holes

Abstract

Let n(k, l, m), k ≤ l ≤ m, be the smallest integer such that any finite planar point set of at least n(k, l, m) points in general position, contains an empty convex k-hole; an empty convex l-hole and an empty convex m-hole, which are all pairwise disjoint. In this paper we prove that n(3, 3, 5) = 12.

A Note on the Minimum Size of a Point Set Containing Three Nonintersecting Empty Convex Polygons

“…Hosono and Urabe in [9] showed n(2, 3, 4) = 9, n(2, 3, 5) = 11, n(4, 4, 4) = 16. We showed n(3, 3, 5) = 12 in [17]. We have proved that n(3, 3, 5) = 12 [17], n(4, 4, 5) ≤ 16 [18] and also discuss a disjoint holes problem in preference [19].…”

“…We showed n(3, 3, 5) = 12 in [17]. We have proved that n(3, 3, 5) = 12 [17], n(4, 4, 5) ≤ 16 [18] and also discuss a disjoint holes problem in preference [19]. In this paper, we will continue discussing this problem and prove that n(3, 4, 5) = 13.…”

Two Disjoint 5-Holes in Point Sets

“…For three parameters, most values h(k 1 , k 2 , k 3 ) for k 1 , k 2 , k 3 ≤ 4 and also the values h(2, 3, 5) = 11 and h(3, 3, 5) = 12 are known [HU08,YW15]. Tables 2 and 3 summarize the currently best known bounds for three-parametric values.…”