  This note investigates cover levels of finite sets in the random
interlacements model introduced in [Ann. of Math. (2) 171 (2010) 2039-2087],
that is, the least level such that the set is completely contained in the
random interlacement at that level. It proves that as the cardinality of a set
goes to infinity, the rescaled and recentered cover level tends in distribution
to the Gumbel distribution with cumulative distribution function $\operatorname
{exp}(-\operatorname {exp}(-z))$.
