Given a positive integer $n\in \mathbb{N}$ we define the Hadwiger-Nelson graph $\text{HN}_n$ by

1. $V(\text{HN}_n) = \mathbb{R}^n$;
2. $E(\text{HN}_n) = \{\{v_1, v_2\}: v_1, v_2 \in \mathbb{R}^n \text{ and } |v_1-v_2| = 1\}$.

Main question: Is it true that the clique number $\omega(\text{HN}_n)$ equals $n+1$ for all $n\in \mathbb{N}$?

Side question: An upper bound for the chromatic number $\chi(\text{HN}_2)$ is known, namely $\chi(\text{HN}_2)\leq 7$. Are upper bounds known for $\chi(\text{HN}_n)$ where $n\geq 3$?

Given a positive integer $n\in \mathbb{N}$ we define the Hadwiger-Nelson graph $\text{HN}_n$ by

1. $V(\text{HN}_n) = \mathbb{R}^n$;
2. $E(\text{HN}_n) = \{\{v_1, v_2\}: v_1, v_2 \in \mathbb{R}^n \text{ and } |v_1-v_2| = 1\}$.

Is it true that the clique number $\omega(\text{HN}_n)$ equals $n+1$ for all $n\in \mathbb{N}$?