Existence thresholds and Ramsey properties of random posets

Let ( n ) denote the power set of [ n ], ordered by inclusion, and let ( n , p ) denote the random poset obtained from ( n ) by retaining each element from ( n ) independently at random with probability p and discarding it otherwise. Given any fixed poset F we determine the threshold for the property that ( n , p ) contains F as an induced subposet. We also asymptotically determine the number of copies of a fixed poset F in ( n ) . Finally, we obtain a number of results on the Ramsey properties of the random poset ( n , p ) .