Your formulation is not equivalent. The actual Ramsey theory formulation is something more like...you need to color, either red or blue, all of the edges of a fully connected graph of 6 elements (there are 15 such edges). No matter which coloring you choose (there are 2^15 of these), there will always be a "triangle" between three nodes where the entire triangle is either fully red, or fully blue. If you were to instead restrict yourself to a graph with 5 elements instead of 6, it's possible[1] to color the edges so there's no triangle where all the elements are the same.

As an exercise, try repeating your same argument for 5 colors/blocks, and note that it still works, when it shouldn't.

[1] - https://commons.wikimedia.org/wiki/File:RamseyTheory_K5_no_m...