论文标题
没有少量的联盟锁定家庭中丰富的元素数量
The number of abundant elements in union-closed families without small sets
论文作者
论文摘要
我们让$ \ MATHCAL {F} $为在\ Mathcal {f} $中关闭的有限套件，$ \ emptyset \ not \ not \ in \ Mathcal {f} $，如果元素属于$ \ Mathcal {f} $的一半以上。在此表示法中，古典Frankl的猜想（1979）断言$ \ Mathcal {f} $具有丰富的元素。 Poonen（1992）指出，如果$ \ nathcal {f} $完全具有一个丰富的元素，那么此元素属于$ \ nathcal {f} $的每一集，而Cui and Hu（2019）和Hu（2019）至少会调查$ \ ntarcal {f} $ k $ k $ k $ k $ k $ k $ k $ k $ k berive ins if insc nif insc $ k abif insc $ k abif insc nif insc $ k obles in capl of insc。至少$ k $。 Cui和Hu猜想这是$ k = 2 $的，并询问这是否也适用于$ k = 3 $和$ k> \ frac {n} {2} $，其中$ n $是最大的$ \ nathcal {f} $的大小。 We show that $\mathcal{F}$ has at least $k$ abundant elements if $k \geq n - 3$, and that $\mathcal{F}$ has at least $k - 1$ abundant elements if $k = n - 4$, and we construct a union-closed family with precisely $k - 1$ abundant elements for every $k$ and $n$ satisfying $n - 4 \geq k \ geq 3 $和$ n \ geq 9 $（对于$ k = 3 $和$ n = 8 $）。我们还注意到，$ \ MATHCAL {F} $始终具有至少$ \ min \ {n，2k -n + 1 \} $丰富的元素。另一方面，我们构建了一个与Union Closed家族的家庭，每$ k $和满足$ n \ geq \ max \ {3，5k-4 \} $的每一个精确的两个丰富元素。最后，我们证明了Cui和Hu对$ K = 2 $的猜想位于Frankl的猜想和Poonen的猜想之间。
We let $\mathcal{F}$ be a finite family of sets closed under taking unions and $\emptyset \not \in \mathcal{F}$, and call an element abundant if it belongs to more than half of the sets of $\mathcal{F}$. In this notation, the classical Frankl's conjecture (1979) asserts that $\mathcal{F}$ has an abundant element. As possible strengthenings, Poonen (1992) conjectured that if $\mathcal{F}$ has precisely one abundant element, then this element belongs to each set of $\mathcal{F}$, and Cui and Hu (2019) investigated whether $\mathcal{F}$ has at least $k$ abundant elements if a smallest set of $\mathcal{F}$ is of size at least $k$. Cui and Hu conjectured that this holds for $k = 2$ and asked whether this also holds for the cases $k = 3$ and $k > \frac{n}{2}$ where $n$ is the size of the largest set of $\mathcal{F}$. We show that $\mathcal{F}$ has at least $k$ abundant elements if $k \geq n - 3$, and that $\mathcal{F}$ has at least $k - 1$ abundant elements if $k = n - 4$, and we construct a union-closed family with precisely $k - 1$ abundant elements for every $k$ and $n$ satisfying $n - 4 \geq k \geq 3$ and $n \geq 9$ (and for $k = 3$ and $n = 8$). We also note that $\mathcal{F}$ always has at least $\min \{ n, 2k - n + 1 \}$ abundant elements. On the other hand, we construct a union-closed family with precisely two abundant elements for every $k$ and $n$ satisfying $n \geq \max \{ 3, 5k-4 \}$. Lastly, we show that Cui and Hu's conjecture for $k = 2$ stands between Frankl's conjecture and Poonen's conjecture.
