Existence thresholds and Ramsey properties of random posets. Issue 4 (6th August 2020)

• Record Type: Journal Article
• Authors: Falgas‐Ravry, Victor; Markström, Klas; Treglown, Andrew; Zhao, Yi
• Is Part Of: Random structures & algorithms. Volume 57:Issue 4(2020)
• Page Start: 1097
• Page End: 1133
• Publication Date: 2020-08-06
• Subjects: boolean lattice -- existence thresholds -- Ramsey properties -- random posets; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20952
• Languages: English
• ISSNs: 1042-9832

Abstract

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 ) .