How to evaluate this expected value:
$$\mathbb{E} \left( \smash{\displaystyle\max_{I\in\mathbb{M}}\sum_{i\in I} \xi_i^2 } \right)\le ?,$$
where $\xi_i\overset{ind}{\sim} N(0,1), \mathbb{M}$-arbitrary set, $\vert I\vert\le C$-cardinality of $I$. I would like to obtain such upper bound: $$\mathbb{E} \left( \smash{\displaystyle\max_{I\in\mathbb{M}}\sum_{i\in I} \xi_i^2 } \right)\le K \displaystyle\max_{I\in\mathbb{M}}\vert I\vert\, $$ where $K>0.$ But put $\mathbb{E}$ after $\max$ - is not correctly.