Lower bounds for the Turán densities of daisies

For integers r ⩾ 2 and t ⩾ 2, an r-uniform t-daisy Dt r is a family of 2t t r-element sets of the form {S ∪ T : T ⊂ U, |T| = t} for some sets S, U with |S| = r −t, |U| = 2t and S ∩U = ∅. It was conjectured by Bollob´as, Leader and Malvenuto (and independently by Bukh) that the Tur´an densities of t-daisies satisfy limr→∞ π(Dt r ) = 0 for all t ⩾ 2 (an equivalent conjecture was made independently by Johnson and Talbot). This has become a well-known problem, and it is still open for all values of t. In this paper, we give lower bounds for the Tur´an densities of r-uniform t-daisies. To do so, we introduce (and make some progress on) the following natural problem in additive combinatorics: for integers m ⩾ 2t ⩾ 4, what is the maximum cardinality g(m, t) of a subset R of Z/mZ such that for any x ∈ Z/mZ and any 2t-element subset X of Z/mZ, there are t distinct elements of X whose sum is not in the translate x + R? This is a slice-analogue of an extremal Hilbert cube problem considered by Gunderson and R¨odl as well as Cilleruelo and Tesor