Ramsey-type results for semi-algebraic relations
Author(s)Conlon, David; Fox, Jacob; Pach, Janos; Sudakov, Benny; Suk, Andrew
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For natural numbers d and t there exists a positive C such that if F is a family of n[superscript C] semi-algebraic sets in R[superscript d] 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 4[superscript n] 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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Proceedings of the 29th annual symposium on Symposuim on computational geometry (SoCG '13)
Association for Computing Machinery (ACM)
David Conlon, Jacob Fox, Janos Pach, Benny Sudakov, and Andrew Suk. 2013. Ramsey-type results for semi-algebraic relations. In Proceedings of the twenty-ninth annual symposium on Computational geometry (SoCG '13). ACM, New York, NY, USA, 309-318.