In the early 1980s, Erdős and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph
H
H
is the infimum over all
d
d
for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least
d
d
contains
H
H
. In particular, they raise the questions of determining the uniform Turán densities of
K
4
(
3
)
−
K_4^{(3)-}
and
K
4
(
3
)
K_4^{(3)}
. The former question was solved only recently by Glebov, Král’, and Volec [Israel J. Math. 211 (2016), pp. 349–366] and Reiher, Rödl, and Schacht [J. Eur. Math. Soc. 20 (2018), pp. 1139–1159], while the latter still remains open for almost 40 years. In addition to
K
4
(
3
)
−
K_4^{(3)-}
, the only
3
3
-uniform hypergraphs whose uniform Turán density is known are those with zero uniform Turán density classified by Reiher, Rödl and Schacht [J. London Math. Soc. 97 (2018), pp. 77–97] and a specific family with uniform Turán density equal to
1
/
27
1/27
.
We develop new tools for embedding hypergraphs in host hypergraphs with positive uniform density and apply them to completely determine the uniform Turán density of a fundamental family of
3
3
-uniform hypergraphs, namely tight cycles
C
ℓ
(
3
)
C_\ell ^{(3)}
. The uniform Turán density of
C
ℓ
(
3
)
C_\ell ^{(3)}
,
ℓ
≥
5
\ell \ge 5
, is equal to
4
/
27
4/27
if
ℓ
\ell
is not divisible by three, and is equal to zero otherwise. The case
ℓ
=
5
\ell =5
resolves a problem suggested by Reiher.