Multiple Distance Ramsey Bounds For Graphs in Euclidean Spaces
For a finite set $A \subset \mathbb{R}_{>0}$ and a finite graph $H$, let $\chi_H(\mathbb{R}^n;A)$ be the minimum number of colors required to color $\mathbb{R}^n$ while avoiding a monochromatic copy of $H$ whose edges have distances in $A$. Extending the graph-copy framework of Axenovich, Liu, and Sagdeev and a multiple distance theorem of Naslund, we prove for any positive integer $m$, \[\chi_H(\mathbb{R}^n;m):=\max_{\substack{A \subseteq \mathbb{R}_{>0} \\ |A|=m}} \chi_H(\mathbb{R}^n;A) \geq \left(\Gamma_{\chi}\sqrt{\frac{m+1}{\Xi(H)}}+o(1)\right)^n.\] Here, $\Gamma_{\chi}$ is a constant and $\Xi(H)$ is an explicit structural parameter that can be substantially smaller than $|V(H)|-1$, thereby recovering Naslund's similar bound for complete graphs and improving the general bound inherited from the corresponding clique for many graph families. Along the way, we construct a weighted strengthening of the semi-diagonal flattening rank theorem of Correia, Sudakov, and Tomon.