The spectral Erd\H{o}s--Ko--Rado problem asks for the largest adjacency-tensor spectral radius of a $t$-intersecting $k$-uniform family. Keevash, Lenz and Mubayi proved that, for fixed $k,t$ and sufficiently large $n$, the unique extremal family is a full $t$-star, and asked whether such a theorem extends to all $n$. Let $\mathcal{A}_r=\{F\in\binom{[n]}k:|F\cap[t+2r]|\ge t+r\}$ be the Frankl families and write $\rho_r$ for their spectral radii. For $2\le t2k-t$, we prove that $\mathcal{A}_0$ is spectrally extremal if and only if $\rho_0\ge\rho_1$; it is unique up to permutation when the inequality is strict, whereas $\mathcal{A}_0$ and $\mathcal{A}_1$ are both extremal at equality. The layerwise pull used in the Ahlswede--Khachatrian cardinality proof is not applicable here: applied directly, it may decrease the spectral radius. Our proof instead pulls all boundary layers simultaneously and applies Perron tail symmetrization. It follows that $\mathcal{A}_0$ is uniquely extremal for $n\ge (t+1)(k-t+1)+\lceil(t+1)\log(t+1)\rceil$; the leading coefficient $t+1$ is best possible for fixed $t$. We also determine all extremal structures for $t=1$ throughout the range $n\ge2k$.
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 $\ell_{r,p}(\mathcal{F})=\sum_{R\in\binom{[n]}r}d_{\mathcal{F}}(R)^p$; at the codegree level, write $co_p(\mathcal{F})=\ell_{k-1,p}(\mathcal{F})$. We introduce a new convex-transference method for degree-power extremal problems and develop it into a reusable input--transfer--rigidity framework independent of any particular set-system problem. 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 Wu--Zhang quadratic theorem to real exponents and answers a problem of Zhou--Yuan throughout the sharp Erd\H{o}s--Ko--Rado range. Second, if $n\geq2k$, a full point-star maximizes $\ell_{r,p}$ for every $1\leq r\leq k-1$ and real $p\geq2$, again with complete equality classification; thus the framework is not confined to codegrees. Third, if $\nu(\mathcal{F})\leq s$ and $n\geq(2s+1)k-s$, then for every real $p\geq1$, $co_p(\mathcal{F})$ is uniquely maximized, up to isomorphism, by all $k$-sets meeting a fixed $s$-set. This removes the integrality restriction on $p$ and replaces previous cubic thresholds or nonexplicit sufficiently-large assumptions with an explicit linear range valid for arbitrary uniformity.
Mengyue Cao, Mei Lu, Haixiang Zhang· 1 citation· ⚡1
Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F_q$, and let $\mathcal F\subsetneq\genfrac{[}{]}{0pt}{}{V}{k}$. The \emph{projective Ore-degree} of $\mathcal F$ is the minimum, over all $k$-subspaces $S\notin\mathcal F$, of the sum of the $\mathcal F$-degrees of the projective points contained in $S$. We prove sharp projective Ore analogues of the vector-space Erd\H{o}s--Ko--Rado and Hilton--Milner theorems. The Ore--Erd\H{o}s--Ko--Rado theorem holds for $n\ge2k+1$, with equality only for a full point-star. For nontrivial intersecting families, we determine the sharp Ore--Hilton--Milner threshold, together with the complete equality classification, when $q\ge3$ and $n\ge2k+1$, or when $q\ge2$ and $n\ge2k+2$. We further determine a sharp projective Ore-degree threshold forcing a direct-sum matching of size $s$ when $s\ge3$ and $n\ge(2s-1)k-s+4$, and derive a multicolour Ramsey consequence.
Mengyue Cao, Mei Lu, Xuyang Yan et al.· 0 citations
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