Semi-algebraic colorings of complete graphs

DOI10.4230/LIPICS.SOCG.2019.36arXiv1505.07429MaRDI QIDQ6262161

János Pach, Andrew Suk, Jacob Fox

Publication date: 27 May 2015

Abstract: 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 $pge3$ and $mge2$, 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. The order of magnitude of this bound is tight. Our proof is based on the Cutting Lemma of Chazelle {em et al.}, and on a Szemer'edi-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 ErdH{o}s and Shelah.

Mathematics Subject Classification ID

Computer graphics; computational geometry (digital and algorithmic aspects) (68U05)

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