发表机构
Institute for Mathematical Sciences, Renmin University of China; Department of Mathematical Sciences, Tsinghua University（中国人民大学数学科学研究所; 清华大学数学科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出离散双矩插值原理，证明全$t$-星等结构在相交族中最大化度幂相关量，扩展了二次界并解决了相关猜想与问题。
AI 中文摘要
对于族$\nt\binom{[n]}k$和$R\nt\binom{[n]}r$，定义$d_{\nt}(R)=|\nt{F}\nt{F}:R\nt{F}|$，$\nt_{r,p}(\nt)=\nt_{R\nt\binom{[n]}r}d_{\nt}(R)^p$；在余度层面记$co_p(\nt)=\nt_{k-1,p}(\nt)$。我们提出离散双矩插值原理，通过二次插值在整数度格上控制$x^p$，将所有实数$p\nt2$的问题简化为前两个下降矩的精确界问题。证明在$n\nt(t+1)(k-t+1)$的精确经典范围内，对所有实数$p\nt2$，全$t$-星在$t$-相交族中最大化$co_p$，并确定所有等号情形。利用Bey的大小敏感二次不等式，将同一框架扩展到所有非平凡度层面：若$\nt$是相交族，$n\nt2k$且$1\tr\tk-1$，则全点星对所有实数$p\nt2$最大化$\nt_{r,p}(\nt)$，且有完整的等号分类。因此，余度定理将精确的Wu--Zhang二次界扩展到所有实数$p\nt2$，完成二次边界等号分类，包含$p=2$时的Brooks--Linz猜想，且对整数指数$p\nt2$，在精确的Erdős--Ko--Rado范围内解决了Zhou--Yuan问题。
英文摘要
For a family $\mathcal F\subseteq\binom{[n]}k$ and $R\in\binom{[n]}r$, let $d_{\mathcal F}(R)=|\{F\in\mathcal F:R\subseteq F\}|$ and $\ell_{r,p}(\mathcal F)=\sum_{R\in\binom{[n]}r}d_{\mathcal F}(R)^p$; write $co_p(\mathcal F)=\ell_{k-1,p}(\mathcal F)$ for the codegree power sum. We introduce a method that uses convexity to extend sharp bounds for degree sums and sums of squared degrees to real powers, while retaining all equality cases. The method bounds $x^p$ by quadratic polynomials or by a continuous function that is linear on each of two intervals. These functions agree with $x^p$ at the degrees of the proposed extremal family, so the argument requires no bounds for sums of higher powers. For families with bounded matching number, we instead use a bound for $\sum_R\max\{d_{\mathcal F}(R)-s,0\}$ together with the degree sum. We give three exact applications. First, a full $t$-star maximizes $co_p$ among $t$-intersecting families for every real $p\geq2$ in the sharp range $n\geq(t+1)(k-t+1)$, with all equality cases determined. This extends the quadratic theorem of Wu and Zhang to real exponents and answers a problem of Zhou and Yuan throughout the sharp Erdős--Ko--Rado range. Second, among intersecting families with $n\geq2k$, a full star maximizes $\ell_{r,p}$ for every $1\leq r\leq k-1$ and real $p\geq2$, again with all equality cases determined. Third, if $ν(\mathcal F)\leq s$ and $n\geq(2s+1)k-s$, then for every real $p\geq1$, $co_p(\mathcal F)$ is uniquely maximized, up to permutation, by all $k$-sets meeting a fixed $s$-set. This removes the integrality restriction on $p$ and replaces previous cubic thresholds or assumptions that $n$ is sufficiently large with an explicit linear range valid for arbitrary $k$.
