Given a fixed hypergraph
H
H
, let
wsat
(
n
,
H
)
\operatorname {wsat}(n,H)
denote the smallest number of edges in an
n
n
-vertex hypergraph
G
G
, with the property that one can sequentially add the edges missing from
G
G
, so that whenever an edge is added, a new copy of
H
H
is created. The study of
wsat
(
n
,
H
)
\operatorname {wsat}(n,H)
was introduced by Bollobás in 1968, and turned out to be one of the most influential topics in extremal combinatorics. While for most
H
H
very little is known regarding
wsat
(
n
,
H
)
\operatorname {wsat}(n,H)
, Alon proved in 1985 that for every graph
H
H
there is a limiting constant
C
H
C_H
so that
wsat
(
n
,
H
)
=
(
C
H
+
o
(
1
)
)
n
\operatorname {wsat}(n,H)=(C_H+o(1))n
. Tuza conjectured in 1992 that Alon’s theorem can be (appropriately) extended to arbitrary
r
r
-uniform hypergraphs. In this paper we prove this conjecture.