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, thecompletegraph 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.

中文翻译：

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半代数着色的 Schur-Erdős 问题

我们考虑一个完整图的边的 m 着色，其中每个颜色类都是半代数定义的，具有有界复杂性。Alon 等人首先研究了 m = 2 的情况，他应用这个框架为几何对象的交集图和计算几何中出现的其他图获得了惊人的强大的 Ramsey 型结果。考虑较大的 m 值是相关的，例如，与由点集确定的不同距离的数量有关的问题。对于 p ≥ 3 和 m ≥ 2，经典的拉姆齐数 R(p; m) 是最小的正整数 n，使得 Kn（n 个顶点上的完全图）的边的任何 m 着色都包含单色 Kp。这是一个长期存在的开放问题，可以追溯到 Schur (1916) 来决定 R(p; m) ≤ 2cm，其中 c = c(p)。我们证明，如果每个颜色类都是用有界复杂度半代数定义的，并且这个界限的数量级是紧的，那么这是真的。我们的证明基于 Chazelle 等人的切割引理，以及多色半代数图的 Szemeredi 型正则引理，这是独立的兴趣。相同的技术用于解决 Erdős 和 Shelah 的更一般的 Ramsey 型问题的半代数变体。