Some results on similar configurations in subsets of ${\mathbb{F}}_q^d$

In this paper, we study problems about the similar configurations in ${\mathbb{F}}_q^d$. Let $G=(V,E)$ be a graph, where $V=\{1,2,\dots ,n\}$ and $E\subseteq \left(\genfrac{}{}{0px}{}{V}{2}\right)$. For a set $\mathcal{E}$ in ${\mathbb{F}}_q^d$, we say that $\mathcal{E}$ contains a pair of $G$ with dilation ratio $r$ if there exist distinct ${\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_n\in \mathcal{E}$ and distinct ${\mathit{y}}_1,{\mathit{y}}_2,\dots ,{\mathit{y}}_n\in \mathcal{E}$ such that $\mathrm{\parallel }{\mathit{y}}_i-{\mathit{y}}_j\mathrm{\parallel }=r\mathrm{\parallel }{\mathit{x}}_i-{\mathit{x}}_j\mathrm{\parallel }\mathrm{\ne }0$ whenever $\{i,j\}\in E$, where $\mathrm{\parallel }\mathit{x}\mathrm{\parallel }:=x_1^2+x_2^2+\cdots +x_d^2$ for $\mathit{x}=(x_1,x_2,\dots ,x_d)\in {\mathbb{F}}_q^d$. We show that if $\mathcal{E}$ has size at least $C_k{q}^{d\mathrm{/}2}$, then $\mathcal{E}$ contains a pair of $k$-stars with dilation ratio $r$, and that if $\mathcal{E}$ has size at least $C\cdot \min \{q^{(2d+1)\mathrm{/}3},\max \{q^3,q^{d\mathrm{/}2}\}\}$, then $\mathcal{E}$ contains a pair of $4$-paths with dilation ratio $r$. Our method is based on enumerative combinatorics and graph theory.