Let $Y$ be a $d$-dimensional standard Gaussian vector, i.e. $Y\sim\mathcal{N}(0,I_d)$. For any subset $S\subset[d]=\{1,2,\cdots,d\}$, let $|S|$ be the size of $S$ and $Y_S$ be the subvector $(Y_i:i\in S)$. This post shows the following: there is some universal constant $c>0$ such that $$\mathbb{E}\max_{|S|=k}||Y_S||_2\leq c\sqrt{k\log\frac{ed}{k}},$$
I would also like to know whether this bound is tight (eg. whether there is some other constant $c'>0$ such that $\mathbb{E}\max_{|S|=k}||Y_S||_2\geq c'\sqrt{k\log\frac{ed}{k}}$). My main concern is, the upper bound proof therein seems to be ignorance of the dependence structure between $Y_S$ and $Y_{S'}$, while this might be a problem for the lower bound?