2 problems

Let $H$ be a finite graph, and write $f_H(r)$ for its growth function, namely the maximum number of vertices in a ball of radius $r$. For graphs $G_1,\dots,G_k$, let…

Let $G$ be the Cayley graph of a nilpotent group with polynomial growth degree $d$, and let $S(r)$ denote the sphere of radius $r$ in $G$. Write $f(r)\lesssim g(r)$ when $f(r)$ is…