ABSTRACT
Most of the theorems discussed in this book address problems on the maximum possible size of a family that possesses some prescribed property P https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429440809/b6425072-5e49-4074-9ee0-5706dabf2404/content/eq5841.tif"/> . In this chapter we present a natural dual problem and will look for the minimum possible size of certain set families. Almost all properties discussed in the book are monotone, i.e. if a family F https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429440809/b6425072-5e49-4074-9ee0-5706dabf2404/content/eq5842.tif"/> possesses P https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429440809/b6425072-5e49-4074-9ee0-5706dabf2404/content/eq5843.tif"/> , then so do all subfamilies of F https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429440809/b6425072-5e49-4074-9ee0-5706dabf2404/content/eq5844.tif"/> . In this case, the empty family has property P https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429440809/b6425072-5e49-4074-9ee0-5706dabf2404/content/eq5845.tif"/> , so asking for the minimal possible size would not be too interesting. Therefore we will be interested in P https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429440809/b6425072-5e49-4074-9ee0-5706dabf2404/content/eq5846.tif"/> -saturated families: families that have property P https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429440809/b6425072-5e49-4074-9ee0-5706dabf2404/content/eq5847.tif"/> , but adding any new set to the family would result in not having property P https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429440809/b6425072-5e49-4074-9ee0-5706dabf2404/content/eq5848.tif"/> anymore. The largest of these families is the same as the largest family that does not have property P https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429440809/b6425072-5e49-4074-9ee0-5706dabf2404/content/eq5849.tif"/> , but this time we are interested in the smallest of these families.
