combinatorics

Projective Ore-Degree Conditions for Intersection Theorems in Vector Spaces

arXiv:2607.25598

summary

The paper introduces a projective Ore-degree measure for families of k‑dimensional subspaces over a finite field and proves sharp analogues of the Erdős–Ko–Rado and Hilton–Milner intersection theorems, including classification of extremal cases and applications to matchings and Ramsey-type results.

Abstract

Let $V$ be an $n$ -dimensional vector space over the finite field $F_{q}$ , and let $F ⊊ [k V]$ . The \emph{projective Ore-degree} of $F$ is the minimum, over all $k$ -subspaces $S \in / F$ , of the sum of the $F$ -degrees of the projective points contained in $S$ . We prove sharp projective Ore analogues of the vector-space ErdÅs--Ko--Rado and Hilton--Milner theorems. The Ore--ErdÅs--Ko--Rado theorem holds for $n \geq 2 k + 1$ , 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 $q \geq 3$ and $n \geq 2 k + 1$ , or when $q \geq 2$ and $n \geq 2 k + 2$ . We further determine a sharp projective Ore-degree threshold forcing a direct-sum matching of size $s$ when $s \geq 3$ and $n \geq (2 s - 1) k - s + 4$ , and derive a multicolour Ramsey consequence.

Topics & keywords

#vector spaces #finite fields #intersecting families #erdos-ko-rado theorem #hilton-milner theorem #projective ore-degreeprojective ore-degreek-subspacespoint-stardirect-sum matchingmulticolour ramsey