Ramsey-type theorems for lines in 3-space

TL;DR

This paper establishes new geometric Ramsey-type theorems for lines in three-dimensional space, providing bounds on clique and independent set sizes in graphs induced by line incidences, with implications for combinatorial geometry.

Contribution

It introduces novel bounds on clique and independent set sizes in line incidence graphs in 3-space, surpassing existing algebraic methods and providing efficient algorithms.

Findings

Intersection graph of n lines has a clique or independent set of size Omega(n^{1/3})

Existence of large subsets of lines stabbed by a single line or avoiding small stabbed subsets

Large subsets of lines lying on a regulus or avoiding regulus configurations

Abstract

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.