I am not very sure how to approach the following questions regarding $\epsilon$-regularity of graphs. Any help would be greatly appreciated!
For a graph G and disjoint vertex sets A and B, given (A, B) is an $\epsilon$-regular pair with density $d$. Also $Y \subset B$ with $(d-\epsilon)^{m-1} |Y| > \epsilon |B|$ where $m \in \mathbb{N}$. Show that $|\{(x_1, x_2, ..., x_m) \in A^{m} : | \bigcap_{i=1}^m \{y \in Y : \{ (x_i, y) \in E(G)\}| \le (d-\epsilon)^{m}|Y|\}| \le m\epsilon |A|^m $
Let $k$ and $k'$ be two positive integers. If (A, B) is an $\epsilon$ regular pair with density d, suppose $d >> \epsilon$ and $|A|=|B|=n$ where n is sufficiently large integer, show that (A, B) has a copy of $\mathcal{K}_{k,k'}$ (the complete bipartite graph on k and k' vertices).
My thoughts: the first question requires us to find a bound for the size of all possible m-tuples of the vertex set A, so that the number of common neighbors for each coordinate vertex of a tuple does not exceed that given value. I think it's wise to start counting the number of neighbors in A of each vertex in Y, and check that it does not exceed the total number of edges. but I'm not sure how to proceed from there.
The second problem only tells us if we take a sufficiently large chunk of vertices from (A, B) it's density essentially remains the same. The problem is trivial when $d \approx 1$ but for any $d \in (0, 1)$ I'm not sure how this observation would lead us to find a copy of $\mathcal{K}_{k,k'}$