In this note we present several results related to conjectures of Erd6s 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, f >~ 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 C ~ such that the set FS~ no the following holds: for each graph G with vertex set { 1,2 ..... n} which contains

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... o copies of the complete graph Kk on k vertices there exists A C_{1 ..... n} such that all finite sums of different elements of A span in G an independent set. An infinite version of this problem was stated by Andr~ts Hajnal, who asked if, for a graph G defined on the set of natural numbers, there exists an infinite set A with the above property. Hajnal's question has been recently answered in the negative by Deuber, Gunderson, Hindman and Strauss in [1]. In the same paper the authors prove also that disjoint and bipartite versions of Hajnal's conjecture hold (see [1] for details). The main result of this note, Theorem 5, asserts that a finite (or, more precisely, 'semi-infinite') version of Hajnal's conjecture remains true as well, which, in particular, settles Erd6s' conjecture in the affirmative. We also provide a simple proof for a bipartite version of Hajnal's question, stated as Theorem 7.