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 Turan-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.

[1]  János Pach,et al.  Ramsey-type results for semi-algebraic relations , 2013, SoCG '13.

[2]  David R. Wood,et al.  On the General Position Subset Selection Problem , 2012, SIAM J. Discret. Math..

[3]  Jean Cardinal,et al.  General position subsets and independent hyperplanes in d-space , 2014, ArXiv.

[4]  Micha Sharir,et al.  Incidences between Points and Lines in Three Dimensions , 2015, SoCG.

[5]  Joel H. Spencer,et al.  Turán's theorem for k-graphs , 1972, Discret. Math..

[6]  Jan Kyncl.  Ramsey-type constructions for arrangements of segments , 2012, Eur. J. Comb..

[7]  Ruixiang Zhang,et al.  Bounds of incidences between points and algebraic curves , 2013, 1308.0861.

[8]  Csaba D. Tóth,et al.  Incidences of not-too-degenerate hyperplanes , 2005, Symposium on Computational Geometry.

[9]  David Conlon,et al.  Erdos-Hajnal-type theorems in hypergraphs , 2011, J. Comb. Theory, Ser. B.

[10]  Micha Sharir,et al.  Incidences between points and lines on a two-dimensional variety , 2015, 1502.01670.

[11]  Larry Guth,et al.  Algebraic methods in discrete analogs of the Kakeya problem , 2008, 0812.1043.

[12]  L. Guth,et al.  On the Erdős distinct distances problem in the plane , 2015 .

[13]  J. Pach,et al.  Erdős-Hajnal-type Results on Intersection Patterns of Geometric Objects , 2008 .

[14]  Gyula Károlyi,et al.  Ramsey-Type Results for Geometric Graphs, I , 1997, Discret. Comput. Geom..

[15]  János Pach,et al.  Ramsey-type results for geometric graphs , 1996, SCG '96.

[16]  Micha Sharir,et al.  Repeated Angles in the Plane and Related Problems , 1992, J. Comb. Theory, Ser. A.

[17]  Endre Szemerédi,et al.  Extremal problems in discrete geometry , 1983, Comb..

[18]  János Pach,et al.  Coloring kk-free intersection graphs of geometric objects in the plane , 2008, SCG '08.

[19]  Leonidas J. Guibas,et al.  Lines in space: Combinatorics and algorithms , 2005, Algorithmica.

[20]  Micha Sharir,et al.  Incidences between Points and Circles in Three and Higher Dimensions , 2005, Discret. Comput. Geom..

[21]  János Pach,et al.  A Ramsey-type result for convex sets , 1994 .

[22]  G. Szekeres,et al.  A combinatorial problem in geometry , 2009 .

[23]  Micha Sharir,et al.  Incidences between points and circles in three and higher dimensions , 2002, SCG '02.

[24]  L. Moser,et al.  AN EXTREMAL PROBLEM IN GRAPH THEORY , 2001 .

[25]  P. Erdös.  On an extremal problem in graph theory , 1970 .

[26]  Micha Sharir.  On Joints in Arrangements of Lines in Space and Related Problems , 1994, J. Comb. Theory, Ser. A.

[27]  Andrew Chi-Chih Yao,et al.  A general approach to d-dimensional geometric queries , 1985, STOC '85.

[28]  Micha Sharir,et al.  On the Number of Incidences Between Points and Curves , 1998, Combinatorics, Probability and Computing.

[29]  Zeev Dvir,et al.  On the Number of Rich Lines in Truly High Dimensional Sets , 2014, SoCG.

[30]  Noga Alon,et al.  Crossing patterns of semi-algebraic sets , 2005, J. Comb. Theory, Ser. A.

[31]  János Pach,et al.  Coloring kk-free intersection graphs of geometric objects in the plane , 2008, SCG '08.