There is a way you can recast Erdős conjecture into a statement about certain inclusions among various compact left and two-sided ideals. Such topological-algebraic statements, and a few combinatorial statements, are proved by Neil Hindman in
Here is a brief sample of one of these topological-algebraic statements. Let $\beta\mathbb{N}$ denote the Stone-Čech compactification of the discrete space $\mathbb{N}$. We can extend the usual addition and multiplication operations on $\mathbb{N}$ to $\beta\mathbb{N}$ to make $(\beta\mathbb{N}, +)$ and $(\beta\mathbb{N}, \cdot)$ both into compact right-topological semigroups. (Right topological semigroup means that $(\beta\mathbb{N}, +)$ and $(\beta\mathbb{N}, \cdot)$ are both semigroups and for all $p$, $q \in \beta\mathbb{N}$ the maps $p \mapsto p+q$ and $p \mapsto p\cdot q$ are continuous.) To see how to actually perform this extension you can read section 3, pgs. 23-28, of this pdf document by Vitaly Bergelson. (However, Bergelson's construction makes $(\beta\mathbb{N}, +)$ into a compact left-topological semigroup.)
Let $L \subseteq \beta\mathbb{N}$. We say $L$ is a left ideal of $(\beta\mathbb{N}, +)$ if $L$ is nonempty and $\beta\mathbb{N} + L \subseteq L$. We define a right ideal of $(\beta\mathbb{N}, +)$ dually, and a (two-sided) ideal is both a left and right ideal. We define left, right, and two-sided ideals of $(\beta\mathbb{N}, \cdot)$ by simply replacing "$+$" with "$\cdot$" above.
Now define the following two subsets of $\beta\mathbb{N}$:
- $\mathcal{AP} = \{p \in \beta\mathbb{N} : A \hbox{ contains APs of arbitrary length for all } A \in p \}$
- $\mathcal{D} = \{p \in \beta\mathbb{N} : \sum_{n\in A} 1/n = \infty \hbox{ for all } A \in p\}$
It's known that $\mathcal{AP}$ is a compact two-sided ideal of $(\beta\mathbb{N}, +)$ and $(\beta\mathbb{N}, \cdot)$, and that $\mathcal{D}$ is a compact left ideal of $(\beta\mathbb{N}, +)$ and $(\beta\mathbb{N}, \cdot)$. Therefore (part of) the main result of Hindman's paper is the
Theorem. The following statements are equivalent.
(a) If $A\subseteq \mathbb{N}$ and $\sum_{n \in A} 1/n = \infty$, then A contains APs of arbitrary length.
(b) $\mathcal{D} \subseteq \mathcal{AP}$.
Of course the point here is that since $\mathcal{D}$ is a left ideal and $\mathcal{AP}$ is a two-sided ideal you would hope to have some nice theorems about inclusion relationships among various compact left, right, and two-sided ideals in $\beta\mathbb{N}$ to lean on. As far as I know, no one has attempted to attack Erdős conjecture from this topological-algebraic viewpoint.
Just to further illustrate the difficulties involved, let me mention that currently there is not even a "purely" topological-algebraic proof of Szemerédi's Theorem yet!
Let $\Delta = \{p \in \beta\mathbb{N} : \overline{d}(A) > 0 \hbox{ for all } A \in p\}$ and let $\Delta^* = \{p \in \beta\mathbb{N} : d^*(A) > 0 \hbox{ for all } A \in p\}$. Here $\overline{d}$ and $d^*$ are the upper asymptotic density and Banach Density. It's known that $\Delta$ is a compact left ideal of $(\beta\mathbb{N},+)$, and $\Delta^*$ is a compact two-sided ideal of $(\beta\mathbb{N}, +)$ and a compact left ideal of $(\beta\mathbb{N}, \cdot)$. In the above paper, Hindman shows that Szemerédi's Theorem is equivalent to each of the inclusions $\Delta \subseteq \mathcal{AP}$ and $\Delta^* \subseteq \mathcal{AP}$.
However, one possible approach to show Szemerédi's Theorem in a topological-algebraic "way" was shown not to work in the paper "Subprincipal Closed Ideals in $\beta\mathbb{N}$" by Dennis Davenport and Hindman. In this paper, they show that $\Delta^*$ intersects every closed ideal of $(\beta\mathbb{N},+)$; but, beyond that, not enough is known about inclusions among compact ideals to prove, algebraically, that $\Delta^* \subseteq \mathcal{AP}$.
