Turán Densities for Daisies and Hypercubes

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. Bollobás, Leader and Malvenuto, and also Bukh, conjectured that the Turán density of the r-daisy tends to zero as r →∞. In this paper we disprove this conjecture. Adapting our construction, we are also able to disprove a folklore conjecture about Turán 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 ≥ 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-Turán densities of hypercubes. We also answer some related questions of Johnson and Talbot.