In 1916, Schur introduced the Ramsey number r(3; m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph K_n, there is a monochromatic copy of K₃. He showed that r(3; m) ≤ O(m!), and a simple construction demonstrates that r(3; m) ≥ 2^Ω(m). An old conjecture of Erdős states that r(3; m) = 2^Θ(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.

中文翻译：

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有界 VC 维意味着 Schur-Erdős 猜想

1916 年，Schur 引入了拉姆齐数r (3; m )，它是最小整数n > 1，使得对于完整图K _n的边的任何m着色，都存在K ₃的单色副本。他证明了r (3; m ) ≤ O ( m !)，并且一个简单的构造证明了r (3; m ) ≥ 2 ^{Ω( m )}。Erdős 的一个古老猜想指出r (3; m ) = 2 ^{Θ( m )}. 在本笔记中，我们证明了具有有界 VC 维的m -colorings的猜想，也就是说，对于m -colorings 的性质，由每个颜色类的顶点的邻域引起的集合系统具有有界.