We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G − I, there is a vertex w ∈ I such that {u, v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NPcomplete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P = NP.