  We consider a random walk on the fully-connected lattice with $N$ sites and
study the time evolution of the number of distinct sites $s$ visited by the
walker on a subset with $n$ sites. A record value $v$ is obtained for $s$ at a
record time $t$ when the walker visits a site of the subset for the first time.
The record time $t$ is a partial covering time when $v<n$ and a total covering
time when $v=n$. The probability distributions for the number of records $s$,
the record value $v$ and the record (covering) time $t$, involving $r$-Stirling
numbers, are obtained using generating function techniques. The mean values,
variances and skewnesses are deduced from the generating functions. In the
scaling limit the probability distributions for $s$ and $v$ lead to the same
Gaussian density. The fluctuations of the record time $t$ are also Gaussian at
partial covering, when $n-v={\mathrm O}(n)$. They are distributed according to
the type-I Gumbel extreme-value distribution at total covering, when $v=n$. A
discrete sequence of generalized Gumbel distributions, indexed by $n-v$, is
obtained at almost total covering, when $n-v={\mathrm O}(1)$. These generalized
Gumbel distributions are crossing over to the Gaussian distribution when $n-v$
increases.
