Výsledky projektu Ramseyova teorie v kombinatorické a výpočetní geometrii

Let S be a subset of R^d with finite positive Lebesgue measure. The Beer index of convexity b(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio c(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate a relationship between these two natural measures of convexity of S.

We show that every subset S of the plane with simply connected components satisfies b(S) <= alpha c(S) for an absolute constant alpha, provided b(S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. asserting that this estimate holds for simple polygons.

We also consider higher-order generalizations of b(S). For 1 <= k <= d, the k-index of convexity b_k(S) of a subset S of R^d is the probability that the convex hull of a (k+1)-tuple of points chosen uniformly independently at random from S is contained in S. We show that for every d >= 2 there is a constant beta(d) > 0 such that every subset S of R^d satisfies b_d(S) <= beta c(S), provided b_d(S) exists. We provide an almost matching lower bound by showing that there is a constant gamma(d) > 0 such that for every epsilon from (0,1] there is a subset S of R^d of Lebesgue measure one satisfying c(S) <= epsilon and b_d(S) >= (gamma epsilon)/log_2(1/epsilon) >= (gamma c(S))/log_2(1/c(S)).

The polynomial partitioning method of Guth and Katz has numerous applications in discrete and computational geometry. It partitions a given n-point set P in R^d using the zero set Z(f) of a suitable d-variate polynomial f. Applications of this result are often complicated by the problem, “What should be done with the points of P lying within Z(f)?” A natural approach is to partition these points with another polynomial and continue further in a similar manner. So far this has been pursued with limited success—several authors managed to construct and apply a second partitioning polynomial, but further progress has been prevented by technical obstacles. We provide a polynomial partitioning method with up to d polynomials in dimension d, which allows for a complete decomposition of the given point set. We apply it to obtain a new algorithm for the semialgebraic range searching problem. Our algorithm has running time bounds similar to a recent algorithm by Agarwal et al., but it is simpler both conceptually and technically. While this paper has been in preparation, Basu and Sombra, as well as Fox, Pach, Sheffer, Suk, and Zahl, obtained results concerning polynomial partitions which overlap with ours to some extent.

A classical conjecture of Erdős and Szekeres states that every set of 2^(k−2)+1 points in the plane in general position contains k points in convex position. In 2006, Peters and Szekeres introduced the following stronger conjecture: every red-blue coloring of the edges of the ordered complete 3-uniform hypergraph on 2^(k−2)+1 vertices contains an ordered k-vertex hypergraph consisting of a red and a blue monotone path that are vertex disjoint except for the common end-vertices.

Applying the state of art SAT solver, we refute the conjecture of Peters and Szekeres. We also apply techniques of Erdős, Tuza, and Valtr to refine the Erdős–Szekeres conjecture in order to tackle it with SAT solvers.