  We consider inference for high-dimensional separately and jointly
exchangeable arrays where the dimensions may be much larger than the sample
sizes. For both exchangeable arrays, we first derive high-dimensional central
limit theorems over the rectangles and subsequently develop novel multiplier
bootstraps with theoretical guarantees. These theoretical results rely on new
technical tools such as Hoeffding-type decomposition and maximal inequalities
for the degenerate components in the Hoeffiding-type decomposition for the
exchangeable arrays. We exhibit applications of our methods to uniform
confidence bands for density estimation under joint exchangeability and penalty
choice for $\ell_1$-penalized regression under separate exchangeability.
Extensive simulations demonstrate precise uniform coverage rates. We illustrate
by constructing uniform confidence bands for international trade network
densities.
