On a question of Erdős concerning cohesive basic sequences

Abstract:

For an arbitrary basic sequence $\mathcal {B}$, set $V(\mathcal {B}) = \{ \# {B_k}|k \in {Z^ + }\}$, where $\# {B_k}$ is the number of pairs (a, b) in $\mathcal {B}$ such that $ab = k$. It is proved that $V(\mathcal {B})$ is unbounded if either $\mathcal {B}$ is cohesive or $\mathcal {B} \not \subset \mathcal {M}$. The set $V(\mathcal {B})$ is determined explicitly in these cases.