Almost Empty Monochromatic Quadrilaterals in Planar Point Sets

Liu, L.; Zhang, Y.

doi:10.1134/s0001434618030082

Cited by 3 publications

(1 citation statement)

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“…We say that an r-hole on S is monochromatic if all its vertices are of the same color, and that it is rainbow 1 if all its vertices are of different colors. Many chromatic variants on problems regarding r-holes in colored points sets have been studied since; see [6,23,1,3,7,21,4,19,8,14,25]. In particular, Aichholzer, Fabila-Monroy, Flores-Peñaloza, Hackl, Huemer, and Urrutia showed that every 2-colored set of n points in the plane determines Ω(n 5/4 ) empty monochromatic triangles [1].…”

Section: Introductionmentioning

confidence: 99%

Empty Rainbow Triangles in $k$-colored Point Sets

Fabila-Monroy¹,

Perz²,

Trujillo-Negrete³

2020

Preprint

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Let S be a set of n points in general position in the plane. Suppose that each point of S has been assigned one of k ≥ 3 possible colors and that there is the same number, m, of points of each color class. A polygon with vertices on S is empty if it does not contain points of S in its interior; and it is rainbow if all its vertices have different colors. Let f (k, m) be the minimum number of empty rainbow triangles determined by S. In this paper we give tight asymptotic bounds for this function. Furthermore, we show that S may not determine an empty rainbow quadrilateral for some arbitrarily large values of k and m.

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