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.

• The paper establishes a sharp upper bound for the ℓ-th largest point-degree using a q-Frankl–Wang approach with Gaussian binomial coefficients, generalizing set-theoretic results.
• It corrects naive q-analogs by proving optimal degree indices through structural decompositions and explicit constructions in finite vector spaces.
• The work reveals key structural obstructions and proposes open problems to guide future studies in extremal combinatorics and related applications in coding theory.

Large Point-Degrees in Intersecting Families of Finite Vector Spaces

Introduction and Context

This paper investigates the order statistics of point-degrees in intersecting families of $k$-dimensional subspaces within finite vector spaces. The central problem is an extension of classical extremal combinatorics, specifically degree bounds in intersecting $k$-uniform families, to the context of vector spaces over finite fields. The study is motivated by recent advances on degree order statistics in set systems, notably the results of Frankl and Wang and of Huang and Rao, which establish sharp bounds on the $\ell$-th largest degree in set-theoretic intersection families (Frankl et al., 19 Nov 2025, Huang et al., 2 Feb 2026). This work translates, strengthens, and corrects these results for the $q$-analog setting.

Let $V$ be an $n$-dimensional vector space over $\mathbb{F}_q$, and $Gr(V,k)$ the set of all $k$-dimensional subspaces. A family $\mathcal{F} \subseteq Gr(V,k)$ is intersecting if $k$0 for all $k$1. For a point $k$2 (a $k$3-dimensional subspace), its degree $k$4 is the number of members of $k$5 containing $k$6. The degrees are ordered, denoted $k$7. The main focus is determining sharp upper bounds for $k$8 at various indices $k$9, and how these bounds generalize earlier work in set theory.

Main Results

$\ell$0-Frankl--Wang Degree Bound

The paper establishes a sharp upper bound for intersecting families in vector spaces, improving upon set-theoretic bounds. Specifically, for $\ell$1, it is shown that:

$\ell$2

where $\ell$3 denotes the square of the number of points in a $\ell$4-dimensional subspace and $\ell$5 is the Gaussian binomial coefficient. This result is strictly stronger than the direct $\ell$6-analog of the set-theoretic $\ell$7 index, since $\ell$8 for $\ell$9, addressing a gap between set and vector-space intersection theory.

Failure and Correction of Naive $q$0-Analogs

The naive $q$1-analog of the $q$2-th degree result from set theory fails due to the existence of vector-space Hilton--Milner constructions with $q$3 points of degree exceeding the standard bound. The paper proves the corrected degree index:

$q$4

for fixed $q$5, large $q$6, and $q$7, relying on structural results by Ihringer and Kupavskii (Ihringer et al., 4 May 2026). This establishes $q$8 as the optimal cutoff.

Degree Bound for Large Indices and Structural Obstructions

For indices corresponding to $q$9 (for $V$0 linear in $V$1), the paper conjectures sharp bounds motivated by saturated Frankl--Hilton--Milner families constructed in (Ihringer et al., 4 May 2026):

$V$2

and proves two necessary conditions for any strict counterexample: such families must exhibit either large diversity outside a point-star or strong clustering of high-degree, noncentral points.

Technical Approaches

The proofs utilize double counting, structural decomposition via core families of points and lines, intricate incidence arguments, and bounds from extremal combinatorics for finite vector spaces. Key technical inputs include:

• The vector-space Hilton--Milner theorem [Blokhuis et al., WXZ2023], providing sharp size bounds for nontrivial-intersecting families.
• Structural theorems by Ihringer and Kupavskii (Ihringer et al., 4 May 2026), allowing reduction to core families controlling the order statistics.
• Explicit constructions and calculation of point-degrees within core examples, demonstrating the necessity of corrected degree indices.
• Elementary but tight Gaussian binomial estimates, underpinning asymptotic bounds for large $V$3 and $V$4.

These tools combine to yield formal reductions showing that the degree bounds follow from extremal nontrivial-intersecting theorems and direct incidence counting, rather than requiring shifting or spectral techniques familiar from the set-theory context.

Implications and Open Problems

The results clarify the landscape of degree order statistics for intersecting families in finite vector spaces, revealing the delicate differences between set-theoretic and vector-space analogs. Practically, they provide precise thresholds for degree indices that can inform combinatorial constructions and coding theory. The theoretically rich corrections are necessary for understanding stability and diversity in intersection theorems, and the sharp constructions establish the limits of analogs for families of subspaces.

Open problems include:

• Determining the exact function $V$5: the maximal number of points attaining degree above the threshold in vector-space intersecting families.
• Establishing the minimal additive threshold $V$6 for validity of the corrected $V$7 theorem across all $V$8.
• Proving localization principles for large-$V$9 degree bounds in the full parameter range, potentially via improved structural decompositions.
• Extending order-statistic degree bounds to higher-dimensional subspaces (not just points), analogous to $n$0-degree EKR-type results [SZ2024].

These questions are likely to drive future developments in vector-space extremal combinatorics, with relevance for both theoretical understanding and applications in finite geometry and coding theory.

Conclusion

The paper provides a rigorous account of degree order statistics in intersecting families of finite vector spaces, correcting naive $n$1-analogs and supplying optimal indices and sharp bounds. Through technical refinement and structural reduction, the results generalize, strengthen, and precisely calibrate the combinatorial theory in the vector space setting. The necessity for corrected indices and the identification of structural obstructions mark substantial theoretical advances, while the closing open problems furnish a roadmap for further research in extremal combinatorics.

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References

See: "Large point-degrees in intersecting families of finite vector spaces" (2606.08709), and associated results in (Frankl et al., 19 Nov 2025, Huang et al., 2 Feb 2026, Ihringer et al., 4 May 2026), [SZ2024], [WXZ2023].