Why does everyone believe, what Rosalind Harker said about RJ. I mean, it is obvious, that she was just part of a plan by RJ to mislead Patrick.
Otherwise you have to explain, why RJ made so much mistakes in that episode.
RJ ordered the skywriter as RJ Solution. So he must have expected that Patrick will go to the skywriter and ask, who ordered that picture and that this will lead to his postbox with a letter from Rosalind it.
If it was not RJ's intention that Patrick finds Rosalind, then he just made a couple of mistakes.