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Ramsey-type theorems for lines in 3-space

par Cardinal, Jean ;Payne, Michael ;Solomon, Noam
Référence Discrete mathematics and theoretical computer science, 18, 3, 14
Publication Publié, 2016

Article révisé par les pairs

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Résumé :

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 the following: • The intersection graph of n lines in R3 has a clique or independent set of size Ω(n1/3). • Every set of n lines in R3 has a subset of √n lines that are all stabbed by one line, or a subset of Ω (n/ log n)1/5 such that no 6-subset is stabbed by one line. • Every set of n lines in general position in R3 has a subset of Ω(n2/3) lines that all lie on a regulus, or a subset of Ω(n1/3) 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 R3 – combined with Turán-type results on independent sets in sparse graphs and hypergraphs. As an intermediate step towards the third result, we also show that for a fixed family of plane algebraic curves with s degrees of freedom, every set of n points in the plane has a subset of Ω(n1−1/s) points incident to a single curve, or a subset of Ω(n1/s) points such that at most s of them lie on a curve. 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.