The paper by David Ellis, Maria-Romina Ivan, and Imre Leader addresses the Turán densities of $r$-daisies and hypercubes. An $r$-daisy is an $r$-uniform hypergraph consisting of six $r$-sets formed by taking the union of an $(r-2)$-set with each of the 2-sets of a disjoint 4-set. The authors disprove the conjecture that the Turán density of the $r$-daisy tends to zero as $r$ approaches infinity. They also provide constructions that disprove a folklore conjecture about the Turán densities of hypercubes. Specifically, they show that the smallest set of vertices of the $n$-dimensional hypercube $Q_n$ that intersects every copy of $Q_d$ has asymptotic density strictly below $1/(d+1)$ for all $d \geq 8$. Additionally, they obtain similar bounds for the edge-Turán densities of hypercubes and answer some related questions posed by Johnson and Talbot, as well as disprove conjectures by Bukh and Griggs-Lu on poset densities. The proofs involve linear-algebraic constructions and blow-ups of these constructions to achieve the desired densities.The paper by David Ellis, Maria-Romina Ivan, and Imre Leader addresses the Turán densities of $r$-daisies and hypercubes. An $r$-daisy is an $r$-uniform hypergraph consisting of six $r$-sets formed by taking the union of an $(r-2)$-set with each of the 2-sets of a disjoint 4-set. The authors disprove the conjecture that the Turán density of the $r$-daisy tends to zero as $r$ approaches infinity. They also provide constructions that disprove a folklore conjecture about the Turán densities of hypercubes. Specifically, they show that the smallest set of vertices of the $n$-dimensional hypercube $Q_n$ that intersects every copy of $Q_d$ has asymptotic density strictly below $1/(d+1)$ for all $d \geq 8$. Additionally, they obtain similar bounds for the edge-Turán densities of hypercubes and answer some related questions posed by Johnson and Talbot, as well as disprove conjectures by Bukh and Griggs-Lu on poset densities. The proofs involve linear-algebraic constructions and blow-ups of these constructions to achieve the desired densities.