The Schur-Erdős problem for semi-algebraic colorings

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 Kn, the complete graph on n vertices, contains a monochromatic Kp. It is a longstanding open problem that goes back to Schur (1916) to decide whether R(p; m) ≤ 2cm, where c = c(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 Szemeredi-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.