# Small set expansion and expanders

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.