Suppose $G = (U \cup V,E)$ is a bipartite graph with $n$ vertices on each side. For sets $X \subseteq U$ and $Y \subseteq V$, let $d(X,Y) = |(X \times Y) \cap E| / (|X||Y|)$ denote the edge density between $X$ and $Y$, and say that the pair $(X,Y)$ is $\varepsilon$-regular if $$ |d(A,B) - d(X,Y)| \lt \varepsilon $$ for every two subsets $A \subseteq X, B \subseteq Y$ such that $|A|, |B| \ge \varepsilon \cdot n$.
Szemerédi's Regularity Lemma says that for every $\varepsilon > 0$ and every bipartite graph $G$, the vertices on each side can be partitioned into classes such that almost all pairs of classes form regular pairs. The number of classes is bounded by a tower function in $1/\varepsilon$. This implies, in particular, that every bipartite graph $G$ contains an $\varepsilon$-regular pair with roughly $\frac{n}{\mathrm{tower}(1/\varepsilon)}$ on each side.
In fact, if all we want to show is that $G$ contains an $\varepsilon$-regular pair, it is possible to do better in terms of the size of the pair. The following paper shows that it is always possible to find an $\varepsilon$-regular pair with $\frac{n}{2^{O(1/\varepsilon)}}$ vertices on each side: https://www.combinatorics.org/ojs/index.php/eljc/article/view/v9i1r1
The crucial feature of these results is that the fraction of vertices on each side of the regular pair (i.e. $\frac{|X|}{n}, \frac{|Y|}{n}$) depends only on $\varepsilon$ and not on $n$. This is very useful in settings where $1/\varepsilon$ is tiny compared to $n$, which is the usual setting where the regularity lemma is applied.
In my application, however, I would like $1/\varepsilon$ to be not much smaller than $n$, say, $\varepsilon \approx 1/\log ^{100} (n)$ or even $\varepsilon \approx 1/n^{0.01}$. In such setting, the above results cannot be applied. Nevertheless, in my application I am willing allow the fraction of vertices in the regular pair to depend on $n$. Are there known variants of the regularity lemma that work in such a setting?
For example, is there a result saying something like "$G$ always contains an $\varepsilon$-regular pair such that each side contains at least $\frac{n}{\mathrm{poly}(1/\varepsilon) \cdot \mathrm{poly}(\log (n))}$ vertices"? I would like the edge density of the regular pair to be not much smaller than the edge density of the original graph.