AI 中文总结
研究非平凡相交族相关的加权独立集多项式，通过容斥原理得到含核部分，利用二级极值定理确定无核余项的精确前置因子，还涉及不同权重情况，给出了\(p\)偏置极值问题的证明及相关结论。
AI 中文摘要
设\(D_n\)是\([n]\)的非空子集上的不相交图，其独立集恰好是\([n]\)上的相交族。我们研究加权独立集多项式\(W(n)=\sum_F\prod_{S\in F}w(S)\)，其中和是对这些族进行的，权重\(w(S)=2^{2^{n - |S|}} - 1\)为双指数形式。通过容斥原理，含核（平凡）部分\(Z_\cap(n)\)是精确的，且\(Z_\cap(n)\sim n\cdot 2^{3^{n - 1}}\)。对于无核余项，我们证明了精确的前置因子\(R(n)=(3/4 + o(1))n\cdot 2^{3^{n - 1} - 2^{n - 1} + 2}\)，由此\(\log_2(Z_\cap(n)/R(n)) = 2^{n - 1} - 2 + \log_2(4/3) + o(1)\)，是一个加法\(o(1)\)，而非仅仅是主导阶的。核心是一个二级极值定理：在除\(n\)个单翻转星之外的无核最大链接系统中，最大权重指数是\(3^{n - 1} - 3\cdot 2^{n - 2} + 6\)，比最大值有固定间隙\(2^{n - 2} - 4\)，且极值者被精确分类。对于权重\(w_B(S)=B^{B^{n - |S|}} - 1\)（\(B\geq2\)为整数）情况类似。组合输入是关于非平凡相交族的\(p\)偏置极值问题：对于所有\(n\geq3\)，\(0 < p\leq1/2\)，\(q = 1 - p\)，\(M_2(n,p)=p - pq^{n - 1} + qp^{n - 1}\)。我们给出了一个简短的自包含的埃尔德什 - 柯 - 拉多证明，其二级刚性为二级极值定理提供支持。
英文摘要
Let $D_n$ be the disjointness graph on the nonempty subsets of $[n]$, whose independent sets are exactly the intersecting families on $[n]$. We study the weighted independent-set polynomial $W(n)=\sum_F\prod_{S\in F}w(S)$, the sum running over these families, for the doubly exponential weight $w(S)=2^{2^{n-|S|}}-1$. The kernel-bearing (trivial) part $Z_\cap(n)$ is exact by inclusion-exclusion and satisfies $Z_\cap(n)\sim n\cdot 2^{3^{n-1}}$. For the kernel-free remainder we prove the exact prefactor $R(n)=(3/4+o(1))n\cdot 2^{3^{n-1}-2^{n-1}+2}$, whence $\log_2(Z_\cap(n)/R(n))=2^{n-1}-2+\log_2(4/3)+o(1)$, an additive $o(1)$, not merely a leading-order one. The engine is a second-level extremal theorem: among kernel-free maximal linked systems other than the $n$ one-flip stars, the largest weight exponent is $3^{n-1}-3\cdot 2^{n-2}+6$, a fixed gap $2^{n-2}-4$ below the maximum, with the extremisers classified exactly. None of this is special to the weight: for $w_B(S)=B^{B^{n-|S|}}-1$ with integer $B\ge 2$ the same stars dominate, the near-extremal families sit a gap $B^{n-2}-B^2$ below, and the prefactor is $1-B^{-B}$. The combinatorial input is the $p$-biased extremal problem for non-trivial intersecting families: $M_2(n,p)=p-pq^{n-1}+qp^{n-1}$ for all $n\ge 3$, $0<p\le 1/2$, $q=1-p$. This first level is essentially known: the extremal family is the Wheel coterie of Peleg and Wool, and at $p=1/Q$ the statement, with its maximiser classification, is the case $r=n$ of Borg's Hilton-Milner theorem for signed sets (2013). We give a short self-contained Erdős-Ko-Rado proof, uniform in real $p\in(0,1/2]$, whose layer-two rigidity feeds the second level. The novelty claimed lies at the second level and in the prefactor, where the classification cannot be read off the layer profile alone: at $n=5$ one profile carries two non-isomorphic types of extremisers.
Comments21 pages, 1 figures, 5 Appendixes
