Existence thresholds and Ramsey properties of random posets. Issue 4 (6th August 2020) - Falgas‐Ravry, Victor; Markström, Klas; Treglown, Andrew; Zhao, Yi
Abstract : Let 𝒫 ( n ) denote the power set of [ n ], ordered by inclusion, and let 𝒫 ( n, p ) denote the random poset obtained from 𝒫 ( n ) by retaining each element from 𝒫 ( n ) independently at random with probability p and discarding it otherwise. Given any fixed poset F we determine the threshold for the property that 𝒫 ( n, p ) contains F as an induced subposet. We also asymptotically determine the number of copies of a fixed poset F in 𝒫 ( n ) . Finally, we obtain a number of results on the Ramsey properties of the random poset 𝒫 ( n, p ) .