Almost Empty Monochromatic Triangles in Planar Point Sets

For positive integers c, s ≥ 1, let M3 (c, s) be the least integer such that any set of at least M3 (c, s) points in the plane, no three on a line and colored with c colors, contains a monochromatic triangle with at most s interior points. The case s = 0 , which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that M3 (1, 0) = 3, M3 (2, 0) = 9, and M3 (c, 0) = ∞, for c ≥ 3. In this paper we extend these results when c ≥ 2 and s ≥ 1. We prove that the least integer λ3 (c) such that M3 (c, λ3 (c)) \u3c ∞ satisfies: ⌊(c-1)/2⌋ ≤ λ3 (c) ≤ c - 2, where c ≥ 2. Moreover, the exact values of M3 (c, s) are determined for small values of c and s. We also conjecture that λ3 (4) = 1, and verify it for sufficiently large Horton sets

Contaje de triángulos en conjuntos de puntos coloreados: un problema de la geometría combinatoria

A classical object of study in combinatorial geometry are sets S of points in the plane. A triangle with vertices from S is called empty if it contains no points of S in its interior. The number of empty triangles depends on the positions of points from S and a burning question is: How many empty triangles are there at least, among all sets S of n points? In order to discard degenerate point configurations, we only consider sets S without three collinear points. In this project, a software has been developed which allows to count the number of empty triangles in a set of n points in the plane. The software permits generation of point sets and their graphical visualization, as well as searching and displaying of optimal point configurations encountered. A point set of a given cardinality is said to be optimal if it contains the minimum number of empty triangles. The objective is to derive bounds on the minimum number of empty triangles by means of experiments realized with our software. The created program also allows to count empty monochromatic triangles in two-colored point sets. A triangle is called monochromatic if its three vertices have the same color. While the first problem has been studied extensively during the last decades, the two-colored version remains to be explored in depth. In this work we also expose our results on the minimum number of empty triangles in (small) two-colored point sets. Also, the treated problem is put in context with related results, such as the Erdös-Szekeres theorem, and a short outline of famous problems which contributed to the rise of combinatorial geometry is presented.Un objeto clásico de estudio en la Geometría combinatoria son conjuntos S de n puntos en el plano. Se dice que un triángulo con vértices en S esta vacío si no contiene puntos de S en su interior. El número de triángulos vacíos depende de cómo se dibujó el conjunto S y una pregunta ardiente es: ¿Cuántos triángulos vacíos hay como mínimo en cada conjunto S de n puntos? Para descartar configuraciones de puntos degeneradas solo se consideran nubes de puntos sin tres puntos colineales

Empty monochromatic simplices

Let S be a k-colored (finite) set of n points in , da parts per thousand yen3, in general position, that is, no (d+1) points of S lie in a common (d-1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3a parts per thousand currency signka parts per thousand currency signd we provide a lower bound of and strengthen this to Omega(n (d-2/3)) for k=2.; On the way we provide various results on triangulations of point sets in . In particular, for any constant dimension da parts per thousand yen3, we prove that every set of n points (n sufficiently large), in general position in , admits a triangulation with at least dn+Omega(logn) simplices.Postprint (author’s final draft

Empty Monochromatic Simplices

Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic $d$-simplices determined by $S$, that is, simplices which have only points from one color class of $S$ as vertices and no points of $S$ in their interior. For $3 \leq k \leq d$ we provide a lower bound of $\Omega(n^{d-k+1+2^{-d}})$ and strengthen this to $\Omega(n^{d-2/3})$ for $k=2$. On the way we provide various results on triangulations of point sets in $\mathbb{R}^d$. In particular, for any constant dimension $d\geq3$, we prove that every set of $n$ points ($n$ sufficiently large), in general position in $\mathbb{R}^d$, admits a triangulation with at least $dn+\Omega(\log n)$ simplices

Erdos-Hajnal-type theorems in hypergraphs

The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0 depends only on the graph H. Except for a few special cases, this conjecture remains wide open. However, it is known that a H-free graph must contain a complete or empty bipartite graph with parts of polynomial size. We prove an analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform hypergraph on n vertices is H-free, for any given H, then it must contain a complete or empty tripartite subgraph with parts of order c(log n)^{1/2 + d(H)}, where d(H) > 0 depends only on H. This improves on the bound of c(log n)^{1/2}, which holds in all 3-uniform hypergraphs, and, up to the value of the constant d(H), is best possible. We also prove that, for k > 3, no analogue of the standard Erdos-Hajnal conjecture can hold in k-uniform hypergraphs. That is, there are k-uniform hypergraphs H and sequences of H-free hypergraphs which do not contain cliques or independent sets of size appreciably larger than one would normally expect.Comment: 15 page

On some partitioning problems for two-colored point sets

Let S be a two-colored set of n points in general position in the plane. We show that S admits at least 2 n 17 pairwise disjoint monochromatic triangles with vertices in S and empty of points of S. We further show that S can be partitioned into 3 n 11 subsets with pairwise disjoint convex hull such that within each subset all but at most one point have the same color. A lower bound on the number of subsets needed in any such partition is also given.Postprint (published version

Drawing the Horton Set in an Integer Grid of Minimum Size

In 1978 Erd\H os asked if every sufficiently large set of points in general position in the plane contains the vertices of a convex $k$-gon, with the additional property that no other point of the set lies in its interior. Shortly after, Horton provided a construction---which is now called the Horton set---with no such $7$-gon. In this paper we show that the Horton set of $n$ points can be realized with integer coordinates of absolute value at most $\frac{1}{2} n^{\frac{1}{2} \log (n/2)}$. We also show that any set of points with integer coordinates combinatorially equivalent (with the same order type) to the Horton set, contains a point with a coordinate of absolute value at least $c \cdot n^{\frac{1}{24}\log (n/2)}$, where $c$ is a positive constant

Monochromatic triangles in three-coloured graphs

In 1959, Goodman determined the minimum number of monochromatic triangles in a complete graph whose edge set is two-coloured. Goodman also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, we determine the minimum number of monochromatic triangles and the colourings which achieve this minimum in a sufficiently large three-coloured complete graph.Comment: Some data needed to verify the proof can be found at http://www.math.cmu.edu/users/jcumming/ckpsty

Red-blue clique partitions and (1-1)-transversals

Motivated by the problem of Gallai on $(1-1)$-transversals of $2$-intervals, it was proved by the authors in 1969 that if the edges of a complete graph $K$ are colored with red and blue (both colors can appear on an edge) so that there is no monochromatic induced $C_4$ and $C_5$ then the vertices of $K$ can be partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani recently strengthened this by showing that it is enough to assume that there is no induced monochromatic $C_4$ and there is no induced $C_5$ in {\em one of the colors}. Here this is strengthened further, it is enough to assume that there is no monochromatic induced $C_4$ and there is no $K_5$ on which both color classes induce a $C_5$. We also answer a question of Kaiser and Rabinovich, giving an example of six $2$-convex sets in the plane such that any three intersect but there is no $(1-1)$-transversal for them