Given two non-negative integers n and s, define $m(n,s)$ to be the maximal number such that in every hypergraph $\mathcal{H}$ on n vertices and with at most $m(n,s)$ edges there is a vertex x such that $|{\mathcal{H}}_x|\ge |E(\mathcal{H})|-s$, where ${\mathcal{H}}_x=\{H\setminus \{x\}:H\in E(\mathcal{H})\}$. This problem has been posed by Füredi and Pach and by Frankl and Tokushige. While the first results were only for specific small values of s, Frankl determined $m(n,2^{d-1}-1)$ for all $d\in \mathbb{N}$ with $d|n$. Subsequently, the goal became to determine $m(n,2^{d-1}-c)$ for larger c. Frankl and Watanabe determined $m(n,2^{d-1}-c)$ for $c\in \{0,2\}$. Other general results were not known so far.

Our main result sheds light on what happens further away from powers of two: We prove that $m(n,2^{d-1}-c)=\frac{n}{d}(2^d-c)$ for $d\ge 4c$ and $d|n$ and give an example showing that this equality does not hold for $c=d$. The other line of research on this problem is to determine $m(n,s)$ for small values of s. In this line, our second result determines $m(n,2^{d-1}-c)$ for $c\in \{3,4\}$. This solves more instances of the problem for small s and in particular solves a conjecture by Frankl and Watanabe.

给定两个非负整数n和s，定义$米（ñ，s）$ 成为最大数，以便在每个超图中 $\mathcal{H}$在n个顶点上，最多$米（ñ，s）$边有一个顶点x这样$|{\mathcal{H}}_X|\ge |E（\mathcal{H}）|-s$， 在哪里 ${\mathcal{H}}_X=\{H\setminus \{X\}：H\in E（\mathcal{H}）\}$。这个问题是由弗雷迪（Füredi）和帕奇（Pach）以及弗兰克（Frankl）和德古吉（Tokuushige）提出的。尽管最初的结果仅针对s的特定小值，但Frankl确定$米（ñ，{\mathrm{2个}}^{d-\mathrm{1个}}-\mathrm{1个}）$ 对全部 $d\in \mathbb{ñ}$ 和 $d|ñ$。随后，目标成为确定$米（ñ，{\mathrm{2个}}^{d-\mathrm{1个}}-C）$对于较大的c。弗兰克（Frankl）和渡边确定$米（ñ，{\mathrm{2个}}^{d-\mathrm{1个}}-C）$ 为了 $C\in \{0，\mathrm{2个}\}$。到目前为止尚无其他一般结果。

我们的主要结果揭示了在远离2的幂的情况下会发生什么：我们证明了 $米（ñ，{\mathrm{2个}}^{d-\mathrm{1个}}-C）=\frac{ñ}{d}（{\mathrm{2个}}^d-C）$ 为了 $d\ge 4C$ 和 $d|ñ$ 并给出一个示例，表明该等式不适用于 $C=d$。关于此问题的另一项研究是确定$米（ñ，s）$对于s的小值。在这一行中，我们的第二个结果确定$米（ñ，{\mathrm{2个}}^{d-\mathrm{1个}}-C）$ 为了 $C\in \{3，4\}$。这解决了小s问题的更多实例，特别是解决了Frankl和Watanabe的猜想。