I'm interested in determining the existence of a permutation group $G\subseteq S_n$ of the following form.

- $G$ is large. Meaning that $G$ have at least $n!/2^{o(n)}$ elements. Equivalently, their index is at most $2^{o(n)}$.
- For some fixed $d$, no subgroup $H$ of $G$ is isomorphic to $S_d$ or to $A_d$.

**Question 1:** How small can $d$ be? For instance, is there a subgroup $G$ of $S_n$ of index $2^{o(n)}$ such that no subgroup of $G$ is isomorphic to $S_{\sqrt{n}}$ or $A_{\sqrt{n}}$?

The second question is whether there are Ramsey-theoretic results relating symmetric groups of size $d$ with large subgroups of $S_n$.

**Question 2:** Are there functions $f,g:\mathbb{N}\rightarrow \mathbb{N}$ such that for every $d<f(n)$, any subgroup $G$ of $S_n$ of order at least $g(d)$ contains a subgroup $H$ isomorphic to $S_d$ (or to $A_n$)?