论文标题
用禁止的彩虹总和的整数着色
Integer colorings with forbidden rainbow sums
论文作者
论文摘要
对于一组正整数$ a \ subseteq [n] $，如果$ r $ a $ a $的颜色，则如果不包含彩虹schur三倍，则不含彩虹总和。在本文中，我们在无汇总的背景下启动了RainbowErdős-Rothchild问题的研究，该问题要求$ [n] $的子集，最大的无彩虹总和$ r $ r $ - 颜色。我们表明，对于$ r = 3 $，间隔$ [n] $是最佳的，而对于$ r \ geq8 $，set $ [\ lfloor n/2 \ rfloor，n] $是最佳的。我们还证明了$ r \ geq4 $的稳定定理。证明依赖于超图容器方法和一些临时稳定性分析。
For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erdős-Rothchild problem in the context of sum-free sets, which asks for the subsets of $[n]$ with the maximum number of rainbow sum-free $r$-colorings. We show that for $r=3$, the interval $[n]$ is optimal, while for $r\geq8$, the set $[\lfloor n/2 \rfloor, n]$ is optimal. We also prove a stability theorem for $r\geq4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.
