Labeled compression schemes for extremal classes

It is a long-standing open problem whether there always exists a compression scheme whose size is of the order of the Vapnik-Chervonienkis (VC) dimension $d$. Recently compression schemes of size exponential in $d$ have been found for any concept class of VC dimension $d$. Previously size $d$ unlabeled compression scheme have been given for maximum classes, which are special concept classes whose size equals an upper bound due to Sauer-Shelah. We consider a natural generalization of the maximum classes called extremal classes. Their definition is based on a generalization of the Sauer-Shelah bound called the Sandwich Theorem which has applications in many areas of combinatorics. The key result of the paper is the construction of a labeled compression scheme for extremal classes of size equal to their VC dimension. We also give a number of open problems concerning the combinatorial structure of extremal classes and the existence of unlabeled compression schemes for them.