论文标题
雏菊的武士密度的下限
Lower bounds for the Turán densities of daisies
论文作者
论文摘要
对于整数，$ r \ geq 3 $和$ t \ geq 2 $，a $ r $ - 均匀$ t $ t $ -daisy $ \ mathcal {d}^t_r $是$ \ binom {2t} {2t} {t} $ r $ r $ r $ - element element的家族， \} $$对于某些集合$ s，u $，带有$ | s | = r-t $，$ | u | = 2t $和$ s \ cap u = \ emptyset $。 Bollobás，Leader和Malvenuto（并独立Bukh）猜想，$ t $ -daisies的turán密度满足$ \ lim \ limits_ {r \ to \ infty}π（\ m nartcal {\ mathcal {d} r {d} _r^t）这已成为一个众所周知的问题，并且对于$ t $的所有值仍然开放。在本文中，我们给出了$ r $ r均匀$ t $ daisies的Turán密度的下限。为此，我们介绍（并取得了一些进展）以下自然问题：对于整数$ m \ geq 2t \ geq 4 $，最大的基数$ g（m，t）$ a子集$ r $ of $ \ mathbb {z}}/m \ mathbb {z} $ \ Mathbb {Z}/M \ Mathbb {Z} $和任何$ 2T $ - 元素子集$ x $ of $ \ m \ m \ m \ m \ m \ m \ m \ mathbb {z} $，$ t $ t $不同的元素的$ x $ of $ x $不在translate $ x+x+r $中？这是Gunderson和Rödl考虑的极端希尔伯特立方体问题的切片，及其概括由Cilleruelo和Tesoro研究。
For integers $r \geq 3$ and $t \geq 2$, an $r$-uniform $t$-daisy $\mathcal{D}^t_r$ is a family of $\binom{2t}{t}$ $r$-element sets of the form $$\{S \cup T \ : T\subset U, \ |T|=t \}$$ for some sets $S,U$ with $|S|=r-t$, $|U|=2t$ and $S \cap U = \emptyset$. It was conjectured by Bollobás, Leader and Malvenuto (and independently Bukh) that the Turán densities of $t$-daisies satisfy $\lim\limits_{r \to \infty} π(\mathcal{D}_r^t) = 0$ 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 $\mathbb{Z}/m\mathbb{Z}$ 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.
