Empty Rainbow Triangles in k -colored Point Sets Ruy Fabila-Monroy, - PowerPoint PPT Presentation

Empty Rainbow Triangles in k -colored Point Sets Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo CINVESTAV, MX and TU Graz, AT EuroCG2020 TU Graz 1 Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Introduction Point set in general position Empty triangle: triangle with no point inside Not empty Empty triangle 2 i Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Introduction k -colored point set: k colors in total For every color there are m points with this color. 2 ii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Introduction Rainbow triangle: triangle, where all three vertices have different colors. Not a rainbow triangle Rainbow triangle, but not empty Empty rainbow triangle 2 iii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Introduction How many empty rainbow triangles are at least in a k -colored point set with n points? All empty rainbow triangles incident to v v 2 iv Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Lower Bound How many empty rainbow triangles are at least in a k -colored point set with n points? ⊲ Take an arbitrary point x . ⊲ Count triangles where x is the rightmost point. ⊲ Sum over all points. ⊲ Happiness 3 i Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Lower Bound Label the colors from 1 to k by their leftmost points. In the 4-colored point set below we have blue as color 1 , red as color 2 , green as color 3 and black as color 4 . p 2 p 3 p 4 p 1 3 ii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Lower Bound Label points of color i from left to right with q 1 up to q k . Count rainbow triangles with q j as rightmost point. p 2 p 3 p 4 = q 1 q 3 p 1 q 2 3 iii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Lower Bound Look at points to the left of q j in angular order. There are at least i − 1 other colors, so at least i − 2 times two points with different colors are neighbored. This gives i − 2 rainbow triangles. p 2 p 3 q 3 p 1 q 2 j − 1 points with color i could lie in these i − 2 triangles. 3 iv Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Lower Bound At least i − j − 1 empty rainbow triangles with q j of color i as rightmost point. 6 k 3 − 1 2 k 2 + 1 Summing over all i and j gives 1 3 k. p 2 p 3 q 3 p 1 q 2 3 v Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Upper bound Main ideas: ⊲ Take a point set on k points with few empty triangles ⊲ Replace each point with a cluster of same colored points ⊲ Have few points for each color that are part of an empty rainbow triangle. 4 i Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Upper bound Main ideas: ⊲ Take a point set on k points with few empty triangles ⊲ Replace each point with a cluster of same colored points ⊲ Have few points for each color that are part of an empty rainbow triangle. 4 ii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Upper bound Main ideas: ⊲ Take a point set on k points with few empty triangles ⊲ Replace each point with a cluster of same colored points ⊲ Have few points for each color that are part of an empty rainbow triangle. 4 iii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Horton set H = H 0 ∪ H 1 where H 0 , H 1 are Horton sets, H 1 is high above H 0 and H 0 is far below H 1 . 5 i Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Horton set H = H 0 ∪ H 1 where H 0 , H 1 are Horton sets, H 1 is high above H 0 and H 0 is far below H 1 . H 1 H 0 5 ii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Horton set H = H 0 ∪ H 1 where H 0 , H 1 are Horton sets, H 1 is high above H 0 and H 0 is far below H 1 . H 1 H 0 5 iii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Horton set Let s be a 01-string, H ∅ = H a Horton set on k points. H s = H s 0 ∪ H s 1 , H s 1 high above H s 0 , H s 0 far below H s 1 . H 101 H 11 H 1 H 10 H 100 H 0 5 iv Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Horton set Number of points in H s : k/ 2 | s | 2 k/ 2 | s | +1 choices for such an edge H s 1 k/ 2 | s | +1 choices for point H s 0 k 2 / 2 | s | empty triangles in H s , not in H s 0 or H s 1 . 5 v Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Colored point set Idea: Take a Horton set on k points and replace each point with a cluster of same colored points. 6 i Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Colored point set Construction of cluster: Every cluster is small and behaves like a point. p 6 p 2 p 8 p 4 p 5 p 1 p 7 p 3 6 ii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Colored point set Order points in angular order around p i . Place a point q 1 between the last point of H 0 and the first point of H 1 with distance ε 1 from p i . H 1 p 6 p 2 p 8 p 4 q 1 p 5 p 1 p 7 p 3 H 0 6 iii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Colored point set Place a point q 2 between the first point of H 0 and the last point of H 1 with distance ε 2 from p i , where ε 2 is much smaller than ε 1 . H 1 p 6 p 2 p 8 p 4 q 1 q 2 p 5 p 1 p 7 p 3 H 0 6 iv Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Colored point set Continue in the same way and place new points closer to p i than the points before. p 6 p 2 p 8 p 4 q 1 H 01 q 2 = q 4 p 5 q 3 p 1 p 7 p 3 H 00 6 v Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Colored point set Place points in every angle, which does not contain a point of the cluster. p 6 p 2 p 8 p 4 q 1 q 2 = q 4 p 5 q 3 p 1 p 7 p 3 6 vi Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Colored point set Place the remaining points sufficiently close to p i . Only the first k added points can be in an empty rainbow triangle. p 6 p 2 p 8 p 4 q 1 q 2 = q 4 p 5 p 5 q 3 p 1 p 7 p 3 6 vii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Colored point set How many such triangles exist? For an empty triangle in H s there are 4 | s | 2 k empty k choices rainbow triangles in colored point set. 2 | s | choices 2 | s | choices 6 viii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Colored point set At most 8( k 2 / 2 2 | s | ) empty triangles in H s . For every such triangle, we have at most 4 | s | 2 k rainbow triangles in P . | s |≤ log 2 ( k ) 8( k 2 / 2 2 | s | )(4 | s | 2 k ) = 192 k 3 + O ( k 2 ) � 6 ix Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Empty rainbow quadrilaterals Does there always exist an empty rainbow quadrilateral? Empty and rainbow like with triangles. The quadrilateral can be convex or concave. not rainbow empty rainbow quadrilaterals 7 i Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Empty rainbow quadrilaterals Does there always exist an empty rainbow quadrilateral? Points set p ′ C 4 4 contains the clusters C 1 , p ′ C 5 3 C 2 , C 3 , C 4 , C 5 and the red points. p ′ C 3 5 p ′ 2 C 1 p ′ C 2 1 7 ii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Empty rainbow quadrilaterals Does there always exist an empty rainbow quadrilateral? C 4 C 5 Not empty C 3 C 1 C 2 7 iii Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Empty rainbow quadrilaterals Does there always exist an empty rainbow quadrilateral? C 4 C 5 Not empty C 3 C 1 C 2 7 iv Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Empty rainbow quadrilaterals Does there always exist an empty rainbow quadrilateral? No C 4 C 5 Not empty C 3 C 1 C 2 7 v Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets

Summary and open problems Number of empty rainbow triangles in a k -colored point set: Θ( k 3 ) , for n ≥ k 2 . Problem 1: What happens if n < k 2 ? There exist point sets without empty rainbow quadrilaterals. Problem 2: Does every sufficiently large k -colored ( k ≥ 4 ) point set with the same number of points in each color class contain an empty rainbow 4 -gon or an empty monochromatic 4 -gon? 8 Ruy Fabila-Monroy, Daniel Perz, Ana Laura Trujillo Empty Rainbow Triangles in k -colored Point Sets