Show that there exists a family with cardinality of $c$, of subsets of $\mathbb{N}$, such that an intersection of any three elements of the family is an infinite (countable) set and the intersection of any four elements of the family is a finite set.
Other than the fact that set of infinite subsets of $\mathbb{N}$ is infinite, I couldn't come up with a way to start. I suppose that such family should be constructed, possibly using set $\mathbb{Q}$, which would then imply that the statement is true for $\mathbb{N}$. Any ideas on how to solve this are welcome.
Edit
Malady suggested in the comments that i try constructing four sets such that the given statement is true for them, which is what I did.
So first I'll define four sets $A=\mathbb{N}$, $B=\{\frac{1}{n}:n\in\mathbb{N}\}$, $C=\{\frac{2}{n}:n\in\mathbb{N},\gcd(2,n)\}$ and $D=\{\frac{3}{n}:n\in\mathbb{N},\gcd(3,n)\}$. Four sets satisfying the conditions from the statement are:
$$A\cup B\cup C$$
$$A\cup B\cup D$$
$$A\cup D\cup C$$
$$D\cup B\cup C$$
This suggests that we can, possibly, construct such family by defining some countable subsets of a countable set and then constructing the elements of the family as some union of those subsets. But I'm not sure how to generalize this so that the family is uncountable.
I will define a family of infinite subsets of the countably infinite set $M=\bigcup_{n\in\mathbb N}M_n$ where $M_n$ is the set of all $3\times n$ matrices of $0$s and $1$s; namely, the family $\{X_a:a\in\{0,1\}^\mathbb N\}$ where $$X_a=\bigcup_{n\in\mathbb N}\{A\in M_n:\text{ some row of }A\text{ is an initial segment of }a\}\subseteq M.$$ If $a,b,c,d$ are four distinct infinite sequences of $0$s and $1$s, then $X_a\cap X_b\cap X_c$ is an infinite subset of $M$ but $X_a\cap X_b\cap X_c\cap X_d$ is finite.
$X_a\cap X_b\cap X_c$ is infinite because, for each $n\in\mathbb N$, there is a $3\times n$ matrix whose first row is an initial segment of $a$, whose second row is an initial segment of $b$, and whose third row is an initial segment of $c$.
For some $N\in\mathbb N$ the $N^\text{th}$ initial segments of $a$, $b$, $c$, and $d$ are all different. Hence $X_a\cap X_b\cap X_c\cap X_d\subseteq\bigcup_{n\lt N}M_n$ which is finite.
More generally, for any positive integer $k$ we can construct a family $\mathcal A\subseteq\mathcal P(\mathbb N)$ with $|\mathcal A|=\mathfrak c$ such that the intersection of any $k$ elements of $\mathcal A$ is infinite while the intersection of any $k+1$ elements of $\mathcal A$ is finite.