Semi-Algebraic Colorings of Complete Graphs

Authors Jacob Fox, János Pach, Andrew Suk

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Author Details

Jacob Fox
  • Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305, USA
János Pach
  • École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland,
  • Rényi Institute, Hungarian Academy of Sciences, Budapest, Hungary
Andrew Suk
  • University of California San Diego, 9500 Gilman Dr, La Jolla, CA 92093, USA

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Jacob Fox, János Pach, and Andrew Suk. Semi-Algebraic Colorings of Complete Graphs. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 36:1-36:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We consider m-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case m = 2 was first studied by Alon et al., who applied this framework to obtain surprisingly strong Ramsey-type results for intersection graphs of geometric objects and for other graphs arising in computational geometry. Considering larger values of m is relevant, e.g., to problems concerning the number of distinct distances determined by a point set. For p >= 3 and m >= 2, the classical Ramsey number R(p;m) is the smallest positive integer n such that any m-coloring of the edges of K_n, the complete graph on n vertices, contains a monochromatic K_p. It is a longstanding open problem that goes back to Schur (1916) to decide whether R(p;m)=2^{O(m)}, for a fixed p. We prove that this is true if each color class is defined semi-algebraically with bounded complexity, and that the order of magnitude of this bound is tight. Our proof is based on the Cutting Lemma of Chazelle et al., and on a Szemerédi-type regularity lemma for multicolored semi-algebraic graphs, which is of independent interest. The same technique is used to address the semi-algebraic variant of a more general Ramsey-type problem of Erdős and Shelah.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Semi-algebraic graphs
  • Ramsey theory
  • regularity lemma


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