Reconfirmation of Two Results on Disjoint Empty Convex Polygons

Wu, Liping; Ding, Ren

doi:10.1007/978-3-540-70666-3_23

Cited by 9 publications

(5 citation statements)

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“…The remaining case of k = 6 was recently solved independently by Gerken [18] and Nicolás [29], who proved that every sufficiently large set of points in general position contains a 6-hole. See [3,4,7,8,10,22,24,25,30,31,32,35,36,37,38,39,40,41] for more on empty convex polygons.…”

Section: Empty Polygonsmentioning

confidence: 99%

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

Abel

Ballinger

Bose

et al. 2010

Graphs and Combinatorics

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We prove the following generalised empty pentagon theorem: for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34 (3):497-506, 2005].2000 Mathematics Subject Classification. 52C10 Erdős problems and related topics of discrete geometry, 05D10 Ramsey theory.Key words and phrases. Erdős-Szekeres Theorem, happy end problem, big line or big clique conjecture, empty quadrilateral, empty pentagon, empty hexagon.

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Section: Empty Polygonsmentioning

confidence: 99%

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

Abel

Ballinger

Bose

et al. 2010

Graphs and Combinatorics

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“…Hosono and Urabe [13] also proved that H(3, 5) = 10 and 12 ≤ H(4, 5) ≤ 14. The results H(3, 4) = 7 and H(4, 5) ≤ 14 were later reconfirmed by Wu and Ding [24]. Very recently, using a Ramsey-type result for disjoint empty convex polygons proved by Aichholzer et al [1], Hosono and Urabe [12] proved that 12 ≤ H(4, 5) ≤ 13, thus improving upon their earlier result.…”

Section: Introductionmentioning

confidence: 74%

On the minimum size of a point set containing a 5-hole and a disjoint 4-hole

Bhattacharya

Das

2011

Studia Scientiarum Mathematicarum Hungarica

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Let H(k; l), k ≦ l denote the smallest integer such that any set of H(k; l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117–122, 2004] proved that 12 ≦ H(4, 5) ≦ 14. Very recently, using a Ramseytype result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481–507, 2007], Hosono and Urabe [Kyoto CGGT, LNCS 4535, 90–100, 2008] improve the upper bound to 13. In this paper, with the help of the same Ramsey-type result, we prove that H(4; 5) = 12.

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“…In [11], Hosono and Urabe also gave n(3, 5) = 10, 12 ≤ n(4, 5) ≤ 14 and 16 ≤ n(5, 5) ≤ 20. The result n(3, 4) = 7 and n(4, 5) ≤ 14 were re-authentication by Wu and Ding [12]. Hosono and Urabe [9] proved n(4, 5) ≤ 13. n(4, 5) = 12 by Bhattacharya and Das was published in [13], who also discussed the convex polygons and pseudo-triangles [14].…”

Section: Introductionmentioning

confidence: 85%