Large point-degrees in intersecting families of finite vector spaces

Abstract

Let \(V\) be an \(n\)-dimensional vector space over the finite field \(\Fq\), and let \(\Gr{V}{k}\) denote the family of all \(k\)-dimensional subspaces of \(V\). A family \(\cF\subseteq\Gr{V}{k}\) is called intersecting if \(\dim(F\cap F')\ge1\) for all \(F,F'\in\cF\). For a point \(P\le V\), let \(d_P(\cF)\) denote the number of members of \(\cF\) that contain \(P\), and order the point-degrees as \(d_1(\cF)\ge d_2(\cF)\ge\cdots\ge d_{\points{n}}(\cF)\), where \(\points{m}=(q^m-1)/(q-1)\) is the number of points in an \(m\)-dimensional subspace. Recent work of Frankl and Wang~\cite{FW2025} and of Huang and Rao~\cite{HR2026} established that for \(k\)-uniform intersecting families \(\cF\subseteq\binom{[n]}{k}\) with \(n\ge2k+1\), the bound \(\binom{n-2}{k-2}\) governs the order statistic \(d_{2k+1}(\cF)\). We prove that every intersecting family \(\cF\subseteq\Gr{V}{k}\) with \(n\ge2k+1\) satisfies \(d_{\points{k}^{2}}(\cF)\le\qbinom{n-2}{k-2}\). The naive \(q\)-analog of the Huang--Rao \((k+2)\)-th degree theorem fails, as a vector-space Hilton--Milner construction has \(\points{k+1}\) points of degree larger than \(\qbinom{n-2}{k-2}\); we prove the corrected bound \(d_{1+\points{k+1}}(\cF)\le\qbinom{n-2}{k-2}\) for fixed \(q\), sufficiently large \(k\), and \(n>3k\), using a structural theorem of Ihringer and Kupavskii~\cite{IK2026}. For larger degree indices \(i\), a saturated Frankl--Hilton--Milner family of Ihringer and Kupavskii identifies the conjectural sharp bound on \(d_{\points{i+1}}(\cF)\), and we prove two necessary conditions that any strict counterexample to this conjecture must satisfy.