  This work studies the maximum possible sign rank of $N \times N$ sign
matrices with a given VC dimension $d$. For $d=1$, this maximum is {three}. For
$d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. For $d >2$, similar but
slightly less accurate statements hold. {The lower bounds improve over previous
ones by Ben-David et al., and the upper bounds are novel.}
  The lower bounds are obtained by probabilistic constructions, using a theorem
of Warren in real algebraic topology. The upper bounds are obtained using a
result of Welzl about spanning trees with low stabbing number, and using the
moment curve.
  The upper bound technique is also used to: (i) provide estimates on the
number of classes of a given VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, and (ii)
design an efficient algorithm that provides an $O(N/\log(N))$ multiplicative
approximation for the sign rank.
  We also observe a general connection between sign rank and spectral gaps
which is based on Forster's argument. Consider the $N \times N$ adjacency
matrix of a $\Delta$ regular graph with a second eigenvalue of absolute value
$\lambda$ and $\Delta \leq N/2$. We show that the sign rank of the signed
version of this matrix is at least $\Delta/\lambda$. We use this connection to
prove the existence of a maximum class $C\subseteq\{\pm 1\}^N$ with VC
dimension $2$ and sign rank $\tilde{\Theta}(N^{1/2})$. This answers a question
of Ben-David et al.~regarding the sign rank of large VC classes. We also
describe limitations of this approach, in the spirit of the Alon-Boppana
theorem.
  We further describe connections to communication complexity, geometry,
learning theory, and combinatorics.
