  For a wide class of Hermitian random matrices, the limit distribution of the
eigenvalues close to the largest one is governed by the Airy point process. In
such ensembles, the limit distribution of the k-th largest eigenvalue is given
in terms of the Airy kernel Fredholm determinant or in terms of Tracy-Widom
formulas involving solutions of the Painlev\'e II equation. Limit distributions
for quantities involving two or more near-extreme eigenvalues, such as the gap
between the k-th and the \ell-th largest eigenvalue or the sum of the k largest
eigenvalues, can be expressed in terms of Fredholm determinants of an Airy
kernel with several discontinuities. We establish simple Tracy-Widom type
expressions for these Fredholm determinants, which involve solutions to systems
of coupled Painlev\'e II equations, and we investigate the asymptotic behavior
of these solutions.
