Most YECs adhere to a model which has the origin of the biosphere at about 6,500 years ago, with a world wide flood at about 4,500 ya. Important to this topic is that, in this model, humans descend from an original pair at 6,500 ya, and go through a flood bottleneck with an effective size of 6 people (Noah's three sons and their wives) at 4,500 ya.
I will show that modern humans have far too much genetic diversity for this model to be correct. We will look at Y-chromosome diversity, and compare what is known about mutation rates to the proposed time scale.
For a start, here's a thirteen generation pedigree study which gives us some idea of the human Y-chromosome base substitution rate.
The first paper derives a point mutation rate of 4 in 13 generations (or approximately 1 for every three generations) for 1/5 of the non-recombining area of the Y-chromosome. The non-recombining area is ~95% of the total chromosome. The Y-chromosome is inherited from father to son, so, in the young earth scenario, we all have the Y-chromosome of Noah, with the only difference being the mutations that have occurred since. The one in three substitution rate puts us all about 60 mutations away from Noah on this 1/5, as 4,500 years equals about 180 generations. That would be about 300 mutations for the entire chromosome.
The second paper searches areas of the Y-chromosome that comprise about one fifth of its total in 36 individuals, and comes out with far too many variations to fit the "Noah" scenario (and far too many to fit the 6,500 year Adam scenario).
Moderators, mindspawn sent me a P.M. indicating that he'd like to continue our discussion on the above subject, but with out the distraction of other participants, which seems reasonable as the topic could easily stray from genetics to other subjects. I suggested the Great Debate forum, so could you please put this narrowed down (Y-chromosome only) version of the topic there.