The paper investigates the Turán densities of daisies and hypercubes. A daisy is an r-uniform hypergraph 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, showing that it is at least approximately 0.29. They also disprove a folklore conjecture about the Turán densities of hypercubes, showing that the asymptotic density of a set of vertices in the n-dimensional hypercube that intersects every copy of the d-dimensional subcube is strictly below 1/(d+1) for d ≥ 8. The authors also provide bounds for the edge-Turán densities of hypercubes and answer related questions posed by Johnson and Talbot. They show that for any d ≥ 8, the asymptotic density of such a set is exponentially small in d. The results have implications for poset densities and the analysis of Boolean functions. The paper also provides constructions and proofs for various bounds on the Turán densities of daisies and hypercubes.The paper investigates the Turán densities of daisies and hypercubes. A daisy is an r-uniform hypergraph 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, showing that it is at least approximately 0.29. They also disprove a folklore conjecture about the Turán densities of hypercubes, showing that the asymptotic density of a set of vertices in the n-dimensional hypercube that intersects every copy of the d-dimensional subcube is strictly below 1/(d+1) for d ≥ 8. The authors also provide bounds for the edge-Turán densities of hypercubes and answer related questions posed by Johnson and Talbot. They show that for any d ≥ 8, the asymptotic density of such a set is exponentially small in d. The results have implications for poset densities and the analysis of Boolean functions. The paper also provides constructions and proofs for various bounds on the Turán densities of daisies and hypercubes.