Independent finite sums in graphs defined on the natural numbers

In this note we present several results related to conjectures of Erdős and Hajnal on the existence of independent sets with good arithmetic properties in a locally sparse graph whose vertices are natural numbers. In particular, we prove that if k, ℓ ≥ 2 and a graph G defined on the natural numbers contains no copies of the complete graph on k vertices, then there exists a subset A ⊆ N such that the set FS⩽ℓ(A) = {∑i∈I ai: I ⊆ N and |I| ⩽ ℓ}, is independent in G, which settles Erdősʹ question in the affirmative.