Random versions of Sperner’s theorem and the Erdő-Ko-Rado theorem
Dániel Gerbner, Balázs Patkós · 2018
In this chapter we address randomized versions of Theorem 1 and Theorem 9. We will consider random subfamilies of 2 [ n ] and ( [ n ] k ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429440809/2714d8f8-ef71-44f4-80bb-22c5d5471b0d/content/eq4060.tif"/> such that every set belongs to the family with probability p independently from all other sets. More precisely Q n ( p ) is the probability space of all families F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429440809/2714d8f8-ef71-44f4-80bb-22c5d5471b0d/content/eq4061.tif"/> ⊆ 2 [ n ] such that ℙ ( Q n ( p ) = F ) = p | F | ( 1 − p ) 2 n − | F | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429440809/2714d8f8-ef71-44f4-80bb-22c5d5471b0d/content/eq4062.tif"/>