General Position Subsets and Independent Hyperplanes in d-Space

Erd\H{o}s asked what is the maximum number $\alpha(n)$ such that every set of $n$ points in the plane with no four on a line contains $\alpha(n)$ points in general position. We consider variants of this question for $d$-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed $d$: - Every set $H$ of $n$ hyperplanes in $\mathbb{R}^d$ contains a subset $S\subseteq H$ of size at least $c \left(n \log n\right)^{1/d}$, for some constant $c=c(d)>0$, such that no cell of the arrangement of $H$ is bounded by hyperplanes of $S$ only. - Every set of $cq^d\log q$ points in $\mathbb{R}^d$, for some constant $c=c(d)>0$, contains a subset of $q$ cohyperplanar points or $q$ points in general position. Two-dimensional versions of the above results were respectively proved by Ackerman et al. [Electronic J. Combinatorics, 2014] and by Payne and Wood [SIAM J. Discrete Math., 2013].Comment: 8 page

Ramsey-type theorems for lines in 3-space

We prove geometric Ramsey-type statements on collections of lines in 3-space. These statements give guarantees on the size of a clique or an independent set in (hyper)graphs induced by incidence relations between lines, points, and reguli in 3-space. Among other things, we prove that: (1) The intersection graph of n lines in R^3 has a clique or independent set of size Omega(n^{1/3}). (2) Every set of n lines in R^3 has a subset of n^{1/2} lines that are all stabbed by one line, or a subset of Omega((n/log n)^{1/5}) such that no 6-subset is stabbed by one line. (3) Every set of n lines in general position in R^3 has a subset of Omega(n^{2/3}) lines that all lie on a regulus, or a subset of Omega(n^{1/3}) lines such that no 4-subset is contained in a regulus. The proofs of these statements all follow from geometric incidence bounds -- such as the Guth-Katz bound on point-line incidences in R^3 -- combined with Tur\'an-type results on independent sets in sparse graphs and hypergraphs. Although similar Ramsey-type statements can be proved using existing generic algebraic frameworks, the lower bounds we get are much larger than what can be obtained with these methods. The proofs directly yield polynomial-time algorithms for finding subsets of the claimed size.Comment: 18 pages including appendi

Motion planning in tori

Let X be a subcomplex of the standard CW-decomposition of the n-dimensional torus. We exhibit an explicit optimal motion planning algorithm for X. This construction is used to calculate the topological complexity of complements of general position arrangements and Eilenberg-Mac Lane spaces associated to right-angled Artin groups.Comment: Results extended to arbitrary subcomplexes of tori. Results on products of even spheres adde