arXiv 2602.016922026-02math.CO

关于交集族第$\ell$-大的度数

On the $\ell$-th largest degree of an intersecting family

National University of Singapore（新加坡国立大学）

Hao Huang, Rui Rao

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AI总结：

研究交集族第2k+1大的度数上限，证明其不超过组合数，并验证Frankl和Wang的猜想，同时对大k时的第k+2大度数给出界。

摘要：

设$\mathcal{F}\subset\binom{[n]}{k}$为一个交集族。对于元素$i\in[n]$，其度数是$\mathcal{F}$中包含i的集合数目。假设度数按$d_{1}\ge d_{2}\ge\cdots\ge d_{n}$排列。Huang和Zhao证明当n>2k时，最小度数满足$d_{n}\le\binom{n-2}{k-2}$，最大值由1-star取得。我们加强这一结果，证明对于n≥2k+1，第2k+1大的度数满足$d_{2k+1}\le\binom{n-2}{k-2}$，从而验证Frankl和Wang的猜想。进一步证明，对于大k和n>12k，第k+2大的度数$d_{k+2}$已不超过$\binom{n-2}{k-2}$。我们开发的技术还为$\varepsilon k \le \ell \le k$和足够大的n>C_{\varepsilon}k时，第$\ell+1$大的度数$d_{\ell+1}$给出紧上界。

英文摘要：

Let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family. For an element $i\in[n]$, the degree of $i$ is the number of sets in $\mathcal{F}$ that contain $i$. Assume that the degrees are ordered as $d_{1}\ge d_{2}\ge\cdots\ge d_{n}$.Huang and Zhao showed that if $n>2k$, then the minimum degree satisfies $d_{n}\le\binom{n-2}{k-2}$, with the maximum attained by the $1$-star. We strengthen this result by proving that for $n\ge 2k+1$, the $(2k+1)$-th largest degree satisfies $d_{2k+1}\le\binom{n-2}{k-2}$, thereby confirming a conjecture of Frankl and Wang. Furthermore, we prove that for large $k$ and $n>12k$, the $(k+2)$-th largest degree $d_{k+2}$ is already at most $\binom{n-2}{k-2}$. The techniques we developed also yield a tight upper bound for the $(\ell+1)$-th largest degree $d_{\ell+1}$ for $\varepsilon k \le \ell \le k$ and sufficiently large $n>C_{\varepsilon} k$.

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