I've just started riding a Specialized road bike I got off ebay. The only problem so far is that the carbon seat post starts slipping down after about 20 minutes. The clamp is not a separate piece but is built into the aluminum frame. I thought I tightened it down pretty good this morning, but same problem. Any solutions? Thanks.

I put some adhesive tape on my slipping post on my Scott Sub 10 and this has helped

You can do a search but with a CF post in particular, you want it devoid of grease. Slipping posts affects many riders. The new Thompson Elite on my steel bike would slip down in spite of its micro serrations to negate slippage. You never want to overtorque the clinch bolt. Instead you want to increase fiction between the post and seat tube I.D. I prefer to not remove the finish of the post...even the surface of the seat post inside the tube. Best approach I found was spraying the post with hairspray...let is dry for a minute or so and install without grease. Best and least invasive approach I have found.
HTH,
George

Watch out for tightening the bolt on a carbon seat post. I have seen a seat post break under the weight of a rider on an over torqued seat post.
Suggestion: Wipe down the seat tube well to remove all traces of grease. Acetone will degrease without leaving a residue. Make sure you know how much post you want showing. Then apply some hairspray to the post and inside the seat tube light application. Insert the post to the desires position and use a torque wrench to apply the correct amount of torque to the bolt. This will work quite well giving the post some grip in the ST without over tightening the bolt.

one solution might be to clean out your seat tube with preferred cleaner & do a light "crosshatch" scruffing with sand paper & clean out tube again with preferred cleaner & wipe post down with isopropyl alcohol..