Density versions of Schur's theorem for ideals generated by submeasures

We characterize ideals of subsets of natural numbers for which some versions of Schur's theorem hold. These are similar to generalizations shown by Bergelson (1986) in [1], and Frankl, Graham and Rödl (1990) in [7]. Additionally, we prove a generalization of an iterated version of Ramsey's theorem.