  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, compression schemes of
size $d$ have been given for maximum classes, which are special concept classes
whose size equals an upper bound due to Sauer-Shelah. We consider a
generalization of maximum classes called extremal classes. Their definition is
based on a powerful generalization of the Sauer-Shelah bound called the
Sandwich Theorem, which has been studied in several areas of combinatorics and
computer science. The key result of the paper is a construction of a sample
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.
