A gap theorem for non-trivial maximal intersecting families and an exact weighted asymptotic

Abstract

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 Erd\H{o}s-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.