Ramsey theory (2nd ed.)
Good splitters for counting points in triangles
Journal of Algorithms
Crossing patterns of semi-algebraic sets
Journal of Combinatorial Theory Series A
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
The Ramsey number R(3, t) has order of magnitude t2/log t
Random Structures & Algorithms
A center transversal theorem for hyperplanes and applications to graph drawing
Proceedings of the twenty-seventh annual symposium on Computational geometry
Higher-order Erdös: Szekeres theorems
Proceedings of the twenty-eighth annual symposium on Computational geometry
An improved bound for the stepping-up lemma
Discrete Applied Mathematics
Hi-index | 0.00 |
For natural numbers d and t there exists a positive C such that if F is a family of nC semi-algebraic sets in Rd of description complexity at most t, then there is a subset F' of F of size $n$ such that either every pair of elements in F' intersect or the elements of F' are pairwise disjoint. This result, which also holds if the intersection relation is replaced by any semi-algebraic relation of bounded description complexity, was proved by Alon, Pach, Pinchasi, Radoicic, and Sharir and improves on a bound of 4n for the family F which follows from a straightforward application of Ramsey's theorem. We extend this semi-algebraic version of Ramsey's theorem to k-ary relations and give matching upper and lower bounds for the corresponding Ramsey function, showing that it grows as a tower of height k-1. This improves on a direct application of Ramsey's theorem by one exponential. We apply this result to obtain new estimates for some geometric Ramsey-type problems relating to order types and one-sided sets of hyperplanes. We also study the off-diagonal case, achieving some partial results.