Document Open Access Logo

Semi-algebraic Ramsey Numbers

Author Andrew Suk

PDF
Thumbnail PDF

File

LIPIcs.SOCG.2015.59.pdf
  • Filesize: 479 kB
  • 15 pages

Document Identifiers

Author Details

Andrew Suk

Cite As Get BibTex

Andrew Suk. Semi-algebraic Ramsey Numbers. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 59-73, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.SOCG.2015.59

Abstract

Given a finite set P of points from R^d, a k-ary semi-algebraic relation E on P is the set of k-tuples of points in P, which is determined by a finite number of polynomial equations and inequalities in kd real variables. The description complexity of such a relation is at most t if the number of polynomials and their degrees are all bounded by t. The Ramsey number R^{d,t}_k(s,n) is the minimum N such that any N-element point set P in R^d equipped with a k-ary semi-algebraic relation E, such that E has complexity at most t, contains s members such that every k-tuple induced by them is in E, or n members such that every k-tuple induced by them is not in E.

We give a new upper bound for R^{d,t}_k(s,n) for k=3 and s fixed. In particular, we show that for fixed integers d,t,s, R^{d,t}_3(s,n)=2^{n^{o(1)}}, establishing a subexponential upper bound on R^{d,t}_3(s,n). This improves the previous bound of 2^{n^C} due to Conlon, Fox, Pach, Sudakov, and Suk, where C is a very large constant depending on d,t, and s. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in R^d. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results.

Subject Classification

Keywords
  • Ramsey theory
  • semi-algebraic relation
  • one-sided hyperplanes
  • Schur numbers

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. H. L. Abbott and D. Hanson. A problem of schur and its generalizations. Acta Arith., 20:175-187, 1972. Google Scholar
  2. H. L. Abbott and L. Moser. Sum-free sets of integers. Acta Arith., 11:392-396, 1966. Google Scholar
  3. P. K. Agarwal and J. Erickson. Optimal partition trees. In In Proc. 26th Ann. ACM Sympos. Comput. Geom., pages 1-10, 2010. Google Scholar
  4. P. K. Agarwal and J. Erickson. Geometric range searching and its relatives. In J. E. Goodman B. Chazelle and R. Pollack, editors, Advances in Dicsrete and Computational Geometry, pages 1-56, 1998. Google Scholar
  5. M. Ajtai, J. Komlós, and E. Szemerédi. A note on ramsey numbers. J. Combin. Theory Ser. A, 29:354-360, 1980. Google Scholar
  6. N. Alon, J. Pach, R. Pinchasi, R. Radoičić, and M. Sharir. Crossing patterns of semi-algebraic sets. J. Combin. Theory Ser. A, 111:310-326, 2005. Google Scholar
  7. S. Basu, R. Pollack, and M. F. Roy. Algorithms in Real Algebraic Geometry. Springer-Verlag, Berlin, 2nd edition edition, 2006. Google Scholar
  8. T. Bohman. The triangle-free process. Adv. Math., 221:1653-1677, 2009. Google Scholar
  9. T. Bohman and P. Keevash. The early evolution of the h-free process. Invent. Math., 181:291-336, 2010. Google Scholar
  10. B. Bukh and M. Matoušek. Erdős-Szekeres-type statements: Ramsey function and decidability in dimension 1. Duke Math. Journal, 63:2243-2270, 2014. Google Scholar
  11. B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir. A singly exponential stratification scheme for real semi-algebraic varieties and its applications. Theor. Comput. Sci., 84:77-105, 1991. Google Scholar
  12. K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, ii. Discrete Comput. Geom., 4:387-421, 1989. Google Scholar
  13. D. Conlon, J. Fox, J. Pach, B. Sudakov, and A. Suk. Ramsey-type results for semi-algebraic relations. Trans. Amer. Math. Soc., 366:5043-5065, 2014. Google Scholar
  14. D. Conlon, J. Fox, and B. Sudakov. Hypergraph ramsey numbers. J. Amer. Math. Soc., 23:247-266, 2010. Google Scholar
  15. V. Dujmović and S. Langerman. A center transversal theorem for hyperplanes and applications to graph drawing. In In Proc. 27th Ann. ACM Sympos. Comput. Geom., pages 117-124, 2011. Google Scholar
  16. M. Eliáš, J. Matoušek, E. Roldán-Pensado, and Z. Safernová. Lower bounds on geometric Ramsey functions. SIAM J. Discrete Math, 28:1960-1970, 2014. Google Scholar
  17. M. Eliáš and J. Matoušek. Higher-order Erdős-Szekeres theorems. Advances in Mathematics, 244:1-15, 2013. Google Scholar
  18. P. Erdős. Some remarks on the theory of graphs. Bull. Amer. Math. Soc., 53:292-294, 1947. Google Scholar
  19. P. Erdős, A. Hajnal, and R. Rado. Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar., 16:93-196, 1965. Google Scholar
  20. P. Erdős and R. Rado. Combinatorial theorems on classifications of subsets of a given set. Proc. London Math. Soc., 3:417-439, 1952. Google Scholar
  21. P. Erdős and G. Szekeres. A combinatorial problem in geometry. Compos. Math., 2:463-470, 1935. Google Scholar
  22. G. Exoo. A Lower Bound for Schur numbers and multicolor Ramsey numbers of K₃. Electronic J. Combinatorics, 1:1-3, 1994. Google Scholar
  23. J. Fox, J. Pach, B. Sudakov, and A. Suk. Erdős-Szekeres-type theorems for monotone paths and convex bodies. Proceedings of the London Mathematical Society, 105:953-982, 2012. Google Scholar
  24. H. Fredricksen and M. Sweet. Symmetric sum-free partitions and lower bounds for schur numbers. Electronic J. Combinatorics, 7:1-9, 2000. Google Scholar
  25. J. H. Kim. The ramsey number r(3,t) has order of magnitude t²/log t. Random Structures Algorithms, 7:173-207, 1995. Google Scholar
  26. V. Koltun. Almost tight upper bounds for vertical decompositions in four dimensions. J. ACM, 51:699-730, 2004. Google Scholar
  27. J. Matoušek. Efficient partition trees. Discrete Comput. Geom., 8:315-334, 1992. Google Scholar
  28. J. Matoušek and E. Welzl. Good splitters for counting points in triangles. J. Algorithms, 13:307-319, 1992. Google Scholar
  29. J. Milnor. On the betti numbers of real varieties. Proc. Amer. Math. Soc., 15:275-280, 1964. Google Scholar
  30. D. Mubayi and A. Suk. A ramsey-type result for geometric 𝓁-hypergraphs. European Journal of Combinatorics, 41:232-241, 2014. Google Scholar
  31. I. Schur. Über die Kongruenz x^m + y^m = z^mmod p. Jahresber. Deutch. Math. Verein., 25:114-117, 1916. Google Scholar
  32. M. J. Steele. Variations on the monotone subsequence theme of Erdős and Szekeres. In D. Aldous, editor, Discrete Probability and Algorithms, IMA Volumes in Mathematics and its Applications, pages 111-131, Berlin, 1995. Springer. Google Scholar
  33. A. Suk. A note on order-type homogeneous point sets. Mathematika, 60:37-42, 2014. Google Scholar
  34. R. Thom. Sur l'homologie des variétés algébriques réelles. In Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pages 255-265, Princeton, N.J., 1965. Princeton University. Google Scholar
  35. H. E. Warren. Lower bounds for approximation by nonlinear manifold. Trans. Amer. Math. Soc., 133:167-178, 1968. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact