Density versions of Schurʹs theorem for ideals generated by submeasures

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