  An $r$-uniform tight cycle of length $\ell>r$ is a hypergraph with vertices
$v_1,\dots,v_\ell$ and edges $\{v_i,v_{i+1},\dots,v_{i+r-1}\}$ (for all $i$),
with the indices taken modulo $\ell$. It was shown by Sudakov and Tomon that
for each fixed $r\geq 3$, an $r$-uniform hypergraph on $n$ vertices which does
not contain a tight cycle of any length has at most $n^{r-1+o(1)}$ hyperedges,
but the best known construction (with the largest number of edges) only gives
$\Omega(n^{r-1})$ edges. In this note we prove that, for each fixed $r\geq 3$,
there are $r$-uniform hypergraphs with $\Omega(n^{r-1}\log n/\log\log n)$ edges
which contain no tight cycles, showing that the $o(1)$ term in the exponent of
the upper bound is necessary.
