9. Let $m$ be a positive integer, and let $T$ denote the set of all subsets of $\{1,2, \ldots, m\}$. Call a subset $S$ of $T \delta$-good if for all $s_{1}, s_{2} \in S, s_{1} \neq s_{2},\left|\Delta\left(s_{1}, s_{2}\right)\right| \geq \delta m$, where $\Delta$ denotes symmetric difference (the symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the largest possible integer $s$ such that there exists an integer $m$ and a $\frac{1024}{2047}$-good set of size $s$.
Answer: 2048
Let $n=|S|$. Let the sets in $S$ be $s_{1}, s_{2}, \ldots, s_{n}$. We bound the sum $\sum_{1 \leq i<j \leq n}\left|\Delta\left(s_{i}, s_{j}\right)\right|$ in two ways. On one hand, by the condition we have the obvious bound
$$
\sum_{1 \leq i<j \leq n}\left|\Delta\left(s_{i}, s_{j}\right)\right| \geq\binom{ n}{2} \delta m
$$

On the other hand, for $1 \leq i \leq m$, let $t_{i}=\left|\left\{1 \leq j \leq n: i \in s_{j}\right\}\right|$. Then it is clear that
$$
\sum_{1 \leq i<j \leq n}\left|\Delta\left(s_{i}, s_{j}\right)\right|=\sum_{k=1}^{m} t_{k}\left(n-t_{k}\right) \leq \frac{n^{2}}{4} m
$$
by AM-GM. Therefore, we get the bound
$$
\binom{n}{2} \delta m \leq \frac{n^{2}}{4} m \Rightarrow n \leq \frac{2 \delta}{2 \delta-1}=2048 .
$$

To give a construction with $n=2048$, take $m=2047$. For the rest of this construction, we will be interpreting the integers $1,2, \ldots, m$ as 11-digit integers in binary. Given this interpretation, define a dot product $x \odot y$ of two positive integers $0 \leq x, y \leq m$ the following way. If $x=\left(x_{1} x_{2} \ldots x_{11}\right)_{2}, y=$ $\left(y_{1} y_{2} \ldots y_{11}\right)_{2}$ in binary, then
$$
x \odot y=\sum x_{i} y_{i} \quad(\bmod 2) \text {. }
$$

Now we can define the sets $s_{1}, s_{2}, \ldots, s_{2048}$. Define
$$
s_{i}=\{1 \leq j \leq m:(i-1) \odot j=1 .\} .
$$

A computation shows that this construction works.

Some notes: here is the motivation behind the construction. We are treating the integers $0,1, \ldots, m$ as the vector space $V=\mathbb{F}_{2}^{11}$, and the sets $s_{i}$ correspond to linear functionals $f_{i}: V \rightarrow \mathbb{F}_{2}$. In particular, the function $f_{i}: V \rightarrow \mathbb{F}_{2}$ is simply defined as $f_{i}(x)=(i-1) \odot x$, which one can easily check to be linear. This construction corresponds to Hadamard matrices of size $2^{11}$.