Quantitative Forms for Combinatorial Partition and Density Theorems

In this article we want to focus on somewhat different quantitative aspects of these partition theorems. In one direction, we will ask not when the desired monochromatic structure must occur, but rather how many monchromatic structures we must have as the size of the set we are partitioning tends to infinity, and the parameters k, l and r are fixed (cf. Theorems 1,2 and 3).