Turán densities for daisies and hypercubes
Abstract An ‐daisy is an ‐uniform hypergraph consisting of the six ‐sets formed by taking the union of an ‐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 ‐daisy tends to zero as . 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 and large , we show that the smallest set of vertices of the ‐dimensional hypercube that intersects every copy of has asymptotic density strictly below , for all . In fact, we show that this asymptotic density is at most , for some constant . As a consequence, we obtain similar bounds for the edge‐Turán 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.