Lower bounds for the Turán densities of daisies

Abstract

For integers $r\ge 3$ and $t\ge 2$, an $r$-uniform $t$-daisy ${\mathcal{D}}_r^t$ is a family of $\left(\genfrac{}{}{0ex}{}{2t}{t}\right)$ $r$-element sets of the form $$\{S\cup T:T\subset U,|T|=t\}$$ for some sets $S,U$ with $|S|=r-t$, $|U|=2t$ and $S\cap U=\varnothing$. It was conjectured by Bollobás, Leader and Malvenuto (and independently Bukh) that the Turán densities of $t$-daisies satisfy $\underset{r\to \infty }{\lim }\pi ({\mathcal{D}}_r^t)=0$ for all $t\ge 2$; 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 $m\ge 2t\ge 4$, what is the maximum cardinality $g(m,t)$ of a subset $R$ of $\mathbb{Z}/m\mathbb{Z}$ such that for any $x\in \mathbb{Z}/m\mathbb{Z}$ and any $2t$-element subset $X$ of $\mathbb{Z}/m\mathbb{Z}$, 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.

arXiv comment11 pages. Minor changes made in response to comments of two anonymous referees