Jean Cardinal ; Michael S. Payne ; Noam Solomon - Ramsey-type theorems for lines in 3-space

dmtcs:1367 - Discrete Mathematics & Theoretical Computer Science, September 19, 2016, Vol. 18 no. 3 -
Ramsey-type theorems for lines in 3-spaceArticle

Authors: Jean Cardinal ; Michael S. Payne ; Noam Solomon

    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 TurĂ¡n-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.

    Volume: Vol. 18 no. 3
    Section: Combinatorics
    Published on: September 19, 2016
    Accepted on: August 29, 2016
    Submitted on: September 19, 2016
    Keywords: Mathematics - Combinatorics,Computer Science - Computational Geometry,05D10, 52C35

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