Demo something. Don't make excuses about why you don't have a demo...demo something, and tell them what you'll change to make it great. The demo is just so they know you can actually build something. Recently there have been quite a few YC companies that have already built something impressive at the time of the interview, so your competition is stiff. But folks with crappy or no demos also make it in, sometimes.
I just think your odds are much better if you demo something. Realize that some of the guys you're up against can produce a working web application that does something useful in a few hours. So, mock something up.
Make a demo. How else can I say it? Demo something!
A bit of advice, orthogonal to whatever PG's answer is:
I too am applying with a large hard problem, but nonetheless I'm going to have a demo. It's a throwaway prototype that, if I tried to use it as production code, would scale to about ten users before the server fell over. See if you can figure out a way to do the same. If you're doing a put-your-favorite-UNIX-command-on-the-web type startup, just make a web-based frontend that actually invokes that UNIX command to do its processing.
I think PG has said that he really likes screencasts, especially on the applications. It's probably a good idea to have both or just the screencast for the application if your demo only works in very specific circumstances.
I would also like to take this opportunity to ask if there is a point for single-person applications.. the faq's say the chances of getting funded are far lower, would this be a non-issue if the applicant is willing to partner with another person/team?
find a cofounder! yc generally funds teams and isn't in the business of matchmaking.
("find a cofounder" is what pg will probably tell you; it's what he told me repeatedly before my interview. it was a big time investment to find the right one, but completely worthwhile. couldn't imagine going through this alone, and it only gets harder to get a cofounder the longer you put it off -- it no longer feels like 'their' idea.)
I just think your odds are much better if you demo something. Realize that some of the guys you're up against can produce a working web application that does something useful in a few hours. So, mock something up.
Make a demo. How else can I say it? Demo something!