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

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

Coloring half-planes and bottomless rectangles

We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almost complete answer for half-planes and pose several open problems. Moreover we give efficient coloring algorithms

More on Decomposing Coverings by Octants

In this note we improve our upper bound given earlier by showing that every 9-fold covering of a point set in the space by finitely many translates of an octant decomposes into two coverings, and our lower bound by a construction for a 4-fold covering that does not decompose into two coverings. The same bounds also hold for coverings of points in $\R^2$ by finitely many homothets or translates of a triangle. We also prove that certain dynamic interval coloring problems are equivalent to the above question

On vertex coloring without monochromatic triangles

We study a certain relaxation of the classic vertex coloring problem, namely, a coloring of vertices of undirected, simple graphs, such that there are no monochromatic triangles. We give the first classification of the problem in terms of classic and parametrized algorithms. Several computational complexity results are also presented, which improve on the previous results found in the literature. We propose the new structural parameter for undirected, simple graphs -- the triangle-free chromatic number $\chi_3$. We bound $\chi_3$ by other known structural parameters. We also present two classes of graphs with interesting coloring properties, that play pivotal role in proving useful observation about our problem. We give/ask several conjectures/questions throughout this paper to encourage new research in the area of graph coloring.Comment: Extended abstrac

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