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- W4387402743 abstract "For integers $r geq 3$ and $t geq 2$, an $r$-uniform {em $t$-daisy} $D^t_r$ is a family of $binom{2t}{t}$ $r$-element sets of the form$${S cup T : Tsubset U, |T|=t }$$for some sets $S,U$ with $|S|=r-t$, $|U|=2t$ and $S cap U = emptyset$. It was conjectured by Bollobás, Leader and Malvenuto (and independently by Bukh) that the Turán densities of $t$-daisies satisfy $limlimits_{r to infty} pi(D_r^t) = 0$ for all $t geq 2$; 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 $m geq 2t geq 4$, what is the maximum cardinality $g(m,t)$ of a subset $R$ of $mathbb{Z}/mmathbb{Z}$ such that for any $x in mathbb{Z}/mmathbb{Z}$ and any $2t$-element subset $X$ of $mathbb{Z}/mmathbb{Z}$, 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 Rődl as well as Cilleruelo and Tesoro." @default.
- W4387402743 created "2023-10-07" @default.
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- W4387402743 date "2023-10-06" @default.
- W4387402743 modified "2023-10-07" @default.
- W4387402743 title "Lower Bounds for the Turán Densities of Daisies" @default.
- W4387402743 doi "https://doi.org/10.37236/11206" @default.
- W4387402743 hasPublicationYear "2023" @default.
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