2022/04/19 by Ellis, David, King, Dylan
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2204.08930
For integers r ≥ 3 and t ≥ 2, an r-uniform t-daisy Dtr is a family of \binom2tt 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 Bollobás, Leader and Malvenuto (and independently Bukh) that the Turán densities of t-daisies satisfy limr → ∞ π(Drt) = 0 for all t ≥ 2; 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 Turán 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 the extremal Hilbert cube problem considered by Gunderson and Rödl and its generalization studied by Cilleruelo and Tesoro.