First-passage percolation is a random growth model defined on
Z
d
\mathbb {Z}^d
using i.i.d. nonnegative weights
(
τ
e
)
(\tau _e)
on the edges. Letting
T
(
x
,
y
)
T(x,y)
be the distance between vertices
x
x
and
y
y
induced by the weights, we study the random ball of radius
t
t
centered at the origin,
B
(
t
)
=
{
x
∈
Z
d
:
T
(
0
,
x
)
≤
t
}
\mathbf {B}(t) = \{x \in \mathbb {Z}^d : T(0,x) \leq t\}
. It is known that for all such
τ
e
\tau _e
, the number of vertices (volume) of
B
(
t
)
\mathbf {B}(t)
is at least order
t
d
t^d
, and under mild conditions on
τ
e
\tau _e
, this volume grows like a deterministic constant times
t
d
t^d
. Defining a hole in
B
(
t
)
\mathbf {B}(t)
to be a bounded component of the complement
B
(
t
)
c
\mathbf {B}(t)^c
, we prove that if
τ
e
\tau _e
is not deterministic, then a.s., for all large
t
t
,
B
(
t
)
\mathbf {B}(t)
has at least
c
t
d
−
1
ct^{d-1}
many holes, and the maximal volume of any hole is at least
c
log
t
c\log t
. Conditionally on the (unproved) uniform curvature assumption, we prove that a.s., for all large
t
t
, the number of holes is at most
(
log
t
)
C
t
d
−
1
(\log t)^C t^{d-1}
, and for
d
=
2
d=2
, no hole in
B
(
t
)
\mathbf {B}(t)
has volume larger than
(
log
t
)
C
(\log t)^C
. Without curvature, we show that no hole has volume larger than
C
t
log
t
Ct \log t
.