Given a graph $G=(V,E)$ on $n$ vertices and $0 \leq \delta \leq 1/2$, we can define the expansion of $G$ over small sets: $h(G,\delta)= \min_{\vert S\vert \leq \delta n } \phi(S)$, with $\phi(S) = \frac{ E(S,\bar{S}) }{ \vert S \vert }$. On the other hand one can ask about the existence of a small graph $G' \subset G$ such that $h(G',\frac{1}{2}) \geq \delta'$ for some $\delta'$.

My question is: say that, for some $\delta$, the graph $G$ obeys $h(G, \delta) \geq \epsilon$, can we then show the existence of $G'\subset G$ a subgraph on $m$ vertices such that $m \in \Omega(\delta n)$, and $h(G',\frac{1}{2}) \in \Omega(\epsilon)$.

In short: when a graph has some degree of small set expansion, can we show the existence of a small sized proper expander? The only resource I could find on this question is this paper https://arxiv.org/pdf/2001.01522.pdf, but the guarantee on the parameters are much weaker.

Given a graph $G=(V,E)$ on $n$ vertices and $0 \leq \delta \leq 1/2$, we can define the expansion of $G$ over small sets:

$$ h(G,\delta)= \min_{\vert S\vert \leq \delta n } \phi(S) \ , $$ with

$$\phi(S) = \frac{ E(S,\bar{S}) }{ \vert S \vert } \ .$$

This naturally defines some notion of "small set" expander. On the other hand a graph is usually said to be expander if $h(G',\frac{1}{2}) \geq \epsilon$ for some sufficiently large $\epsilon$.

My question is: can small set expansion be concentrated into regular expansion? Formally, say that, for some $\delta$, the graph $G$ obeys $h(G, \delta) \geq \epsilon$, does there exist universal constants $c_1,c_2 > 0$ such that there exists a subgraph $G'\subset G$ on $m \geq c_1 \delta n$ vertices, and $h(G',\frac{1}{2}) \geq c_2 \epsilon$ ? I could also settle for degree-depend bounds.

In short: when a graph has some degree of small set expansion, can we show the existence of a small sized proper expander? The only resource I could find on this question is this paper https://arxiv.org/pdf/2001.01522.pdf, but the guarantee on the parameters are much weaker.

Edit: I think I found a way to show that $h(G',\frac{1}{2}) \geq c_2 \epsilon/\log(n)$ is possible.

Given a graph $G=(V,E)$ on $n$ vertices and $0 \leq \delta \leq 1/2$, we can define the expansion of $G$ over small sets: $h(G,\delta)= \min_{\vert S\vert \leq \delta n } \phi(S)$, with $\phi(S) = \frac{ E(S,\bar{S}) }{ \vert S \vert }$. On the other hand one can ask about the existence of a small graph $G' \subset G$ such that $h(G',\frac{1}{2}) \geq \delta'$ for some $\delta'$.

My question is: say that, for some $\delta$, the graph $G$ obeys $h(G, \delta) \geq \epsilon$, can we then show the existence of $G'\subset G$ a subgraph on $m$ vertices such that $m \in \Omega(\delta n)$, and $h(G',\frac{1}{2}) \in \Omega(\epsilon)$.

In short: when a graph has some degree of small set expansion, can we show the existence of a small sized proper expander? The only resource I could find on this question is this paper https://arxiv.org/pdf/2001.01522.pdf, but only works if $h(G,\frac{1}{2}) \leq \frac{\delta}{2}$.

Given a graph $G=(V,E)$ on $n$ vertices and $0 \leq \delta \leq 1/2$, we can define the expansion of $G$ over small sets: $h(G,\delta)= \min_{\vert S\vert \leq \delta n } \phi(S)$, with $\phi(S) = \frac{ E(S,\bar{S}) }{ \vert S \vert }$. On the other hand one can ask about the existence of a small graph $G' \subset G$ such that $h(G',\frac{1}{2}) \geq \delta'$ for some $\delta'$.

My question is: say that, for some $\delta$, the graph $G$ obeys $h(G, \delta) \geq \epsilon$, can we then show the existence of $G'\subset G$ a subgraph on $m$ vertices such that $m \in \Omega(\delta n)$, and $h(G',\frac{1}{2}) \in \Omega(\epsilon)$.

In short: when a graph has some degree of small set expansion, can we show the existence of a small sized proper expander? The only resource I could find on this question is this paper https://arxiv.org/pdf/2001.01522.pdf, but the guarantee on the parameters are much weaker.