Let $X$ be a set. We say that $\mathcal A\subseteq \mathcal P(X)$ is an independent family (IF) if and only if for any $m,n\in \mathbb N$, and mutually distinct sets $x_0,\ldots,x_m,y_0,\ldots,y_m\in\mathcal A$, $$ \left|\bigcap_{i<m}x_i\backslash \bigcup_{j<n}y_j\right|\neq 0. $$
I am now trying to prove the following theorem: Let $X$ be a set, with $|X|=\kappa$. Then there exists an independent family $\mathcal A\in\mathcal P(X)$ such that $|\mathcal A|=2^\kappa$. The following is an incomplete proof which I recall vaguely from my memory:
Define $Y=\{(F,\mathcal H)|F\subseteq X,|F|<\infty,\mathcal H\subseteq\mathcal P(F)\}$, and for $A\subseteq X$, let $Y_A=\{(F,\mathcal H)\in Y|A\cap F\in \mathcal H\}$. Also, let $\mathcal Y=\{Y_E:E\in \mathcal P(X)\}$. Presumably $|\mathcal Y|=2^\kappa$.
We know prove that $\mathcal Y$ is an independent family of $Y$. Suppose $Y_{A_1},\ldots,Y_{A_n},Y_{A_{n+1}},\ldots,Y_{A_{m+n}}\in \mathcal Y$ are mutually distinct, and we need to prove that $Y_{A_1}\cap\ldots\cap Y_{A_n}\cap Y_{A_{n+1}}^C\cap \ldots \cap Y_{A_{m+n}}^C\neq\emptyset$. To do this, we can construct $(F,\mathcal H)\in Y$, such that $A_i\cap F\in \mathcal H$ for $1\leq i\leq n$ and $A_j\cap F\notin \mathcal H$ for $n+1\leq j\leq m$. Choose $x(i,j)\in X$ to be in exactly one of $A_i,A_j$. Let $F=\{x(i,j)|1\leq i<j\leq m+n\}$, and let $\mathcal H=\{A_i\cap F|1\leq i\leq n\}$. Clearly, $(F,\mathcal H)\in Y$, and $A_i\cap F\in \mathcal H$ for $1\leq i\leq n$. Also, for $n+1\leq i\leq n+m$, $A_i\cap F \notin \mathcal H$, because if $A_i\cap F \in \mathcal H$, then there exist $1\leq j\leq n$ such that $A_i\cap F=A_j\cap F$; however, we have $x(i,j)\in F$ that lies in exactly one of $A_i, A_j$, a contradiction.
Finally, note that $|Y|=|X|$, which completes the proof.
Now there are several doubtful things here. Firstly, I don't know how to prove the claim $|\mathcal Y|=2^\kappa$. I can see clearly that $\mathcal Y\leq2^\kappa$, but why equality should be achieved?
Secondly, the last step is incomplete. The fact that $|X|=|Y|$ help us identify $Y$ to $X$, but I am not sure if it helps to identify $\mathcal P(Y)$ to $\mathcal P(X)$.
I kind of know what's going on here, but when I try to write it down, I get bogged down into details and just cannot finish writing it.
Also, can anyone explain intuitively how we come up with the sets $Y$ and $\mathcal Y$? It appears to me that they come from nowhere.