There have been many proposals over the time for why four dimensions (or four _large_ dimensions!) might be singled out by theory or by some dynamics. Just recently one could see for instance the following arXiv preprint

Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, _Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions_ ([arXiv:1108.1540](https://arxiv.org/abs/1108.1540))

claiming that computer simulations of a certain description of nonperturbative string theory show that exactly 3+1 dimensions dynamically become macroscopic in this theory. Similar statements have been made every now and then. One needs to be a bit careful.


Notice that your statement about the role of Calabi-Yau compactification in string theory is not correct. There is nothing in the theory itself that singles out spacetimes that contain a 6-dimensional Calabi-Yau space as a factor (locally). Rather, a little computation shows that IF one assumes the background geometry to be of this form, with the Riemannian size of the CY factor very small, then it follows that the [effective QFT](https://ncatlab.org/nlab/show/effective+quantum+field+theoryeffective+quantum+field+theory) after the [Kaluza-Klein compactification](https://ncatlab.org/nlab/show/Kaluza-Klein+mechanism) in the remaining four dimensions has precisely one [global supersymmetry](https://ncatlab.org/nlab/show/supersymmetry) at intermediate energy scales. Until very recently, it was widely expected that this is a property that corresponds to our observed world, and that was the only reason for considering these backgrounds. This may be changing as we speak: new experimental results from the LHC these days increasingly disfavor this prejudice. You may find this related blog discussion here useful: _[Local and global supersymmetry](https://golem.ph.utexas.edu/category/2011/07/local_and_global_supersymmetry.html)_