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Semi-Algebraic Colorings of Complete Graphs Fox, Jacob Pach, János Suk, Andrew Semi-algebraic graphs Ramsey theory regularity lemma 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jacob Fox and János Pach and Andrew Suk 2019 Is Part Of LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2019.36 urn:nbn:de:0030-drops-104401 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.36 eng https://creativecommons.org/licenses/by/3.0/legalcode