  An $r$-daisy is an $r$-uniform hypergraph consisting of the six $r$-sets
formed by taking the union of an $(r-2)$-set with each of the 2-sets of a
disjoint 4-set. Bollob\'as, Leader and Malvenuto, and also Bukh, conjectured
that the Tur\'an density of the $r$-daisy tends to zero as $r \to \infty$. In
this paper we disprove this conjecture.
  Adapting our construction, we are also able to disprove a folklore conjecture
about Tur\'an densities of hypercubes. For fixed $d$ and large $n$, we show
that the smallest set of vertices of the $n$-dimensional hypercube $Q_n$ that
meets every copy of $Q_d$ has asymptotic density strictly below $1/(d+1)$, for
all $d \geq 8$. In fact, we show that this asymptotic density is at most $c^d$,
for some constant $c<1$. As a consequence, we obtain similar bounds for the
edge-Tur\'an densities of hypercubes. We also answer some related questions of
Johnson and Talbot, and disprove a conjecture made by Bukh and by Griggs and Lu
on poset densities.
