2026/07/29 by Mengyu Cao, Mei Lu, Xuyang Yan +1
Mathematics · #math.CO #msc:05C65 #msc:05D05
arxiv created 2026/07/29 · arxiv updated 2026/07/30
Let V be an n-dimensional vector space over the finite field \mathbb Fq, and let \mathcal F\subsetneq\genfrac[]0ptVk. The projective Ore-degree of \mathcal F is the minimum, over all k-subspaces S∉\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 Erdős--Ko--Rado and Hilton--Milner theorems. The Ore--Erdős--Ko--Rado theorem holds for n≥2k+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≥3 and n≥2k+1, or when q≥2 and n≥2k+2. We further determine a sharp projective Ore-degree threshold forcing a direct-sum matching of size s when s≥3 and n≥(2s-1)k-s+4, and derive a multicolour Ramsey consequence.