Does there exist an uncountable collection $\Lambda$ of infinite subsets of the set of natural numbers such that (i) any two distinct subsets in the collection have a finite intersection and (ii) the sum of the reciprocals is divergent for each $A \in \Lambda$?



Sets of natural numbers with finite intersections and divergent sums of reciprocals
This can be done by a fairly simple diagonalization argument. Call a subset of $\mathbb{N}$ large if its sum of reciprocals is infinite. It suffices to prove the following (for we can then construct an uncountable family by induction). Let $\Lambda$ be a countable collection of large sets such that for any finite subset $F\subseteq\Lambda$, $\mathbb{N}\setminus\bigcup F$ is large. Then there exists a large set $C$ that has finite intersection with every element of $\Lambda$ such that for every finite subset $F\subseteq\Lambda\cup\{C\}$, $\mathbb{N}\setminus\bigcup F$ is large.


To prove this, let $\{A\_n\}$ be an enumeration of $\Lambda$, and let $B\_n=\bigcup\_{i=0}^n A\_i$; we want to construct a large set $C$ such that $C\cap B\_n$ is finite and $\mathbb{N}\setminus (C\cup B\_n)$ is large for all $n$. We define the characteristic function $1\_C:\mathbb{N}\to\{0,1\}$ of $B$ by the following process: First, make $1\_C$ be $1$ at finitely many points such that the reciprocal sum of $C$ is at least $1$. Then, add finitely many $0$s such that the reciprocal sum of $\mathbb{N}\setminus (C\cup B\_0)$ is at least $1$. Now declare $1\_C$ to be $0$ on all points of $B\_0$ where we have not yet defined it. Now, add finitely many $1$s such that the reciprocal sum of $C$ is at least $2$. Now add finitely many $0$s such that the reciprocal sum of $\mathbb{N}\setminus (C\cup B\_1)$ is at least $2$. Now declare $1\_C$ to be $0$ at all points of $B\_1$ where we have not yet defined it. Now add finitely many $1$s such that the reciprocal sum of $C$ is at least $3$. And so on.


