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A product theorem for r-cross intersecting families of subspaces
Abstract
Let V be an n-dimensional vector space over a finite field of order q. Let r≥ 3, (r−1)n≥ rk and let ℱ1,…,ℱr⊂ genfrac[]0ptVk, where genfrac[]0ptVk denotes the set of k-dimensional subspaces of V. Suppose that F1∩⋯∩ Fr≠{0} holds for all Fi∈ℱi, 1≤ i≤ r. Then we show that ∏i=1r|ℱi|≤genfrac[]0ptn−1k−1, provided n−k is sufficiently large for fixed q and r. Moreover, equality holds if and only if there is a common line L such that every family ℱi consists of all k-dimensional subspaces containing the line L. One of the main tools of the proof is a junta theorem concerning intersecting linear maps obtained by Ellis, Kindler, and Lifshitz.
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BibTeX
@misc{shimizu2026product,
title = {{A product theorem for $r$-cross intersecting families of subspaces}},
author = {Toshihiro Shimizu and Norihide Tokushige},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28013},
eprint = {2608.28013},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28013},
}
RIS
TY - UNPB
TI - A product theorem for r-cross intersecting families of subspaces
AU - Toshihiro Shimizu
AU - Norihide Tokushige
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28013
UR - https://arxiv.org/abs/2608.28013
AB - Let V be an n-dimensional vector space over a finite field of order q. Let r≥ 3, (r−1)n≥ rk and let ℱ_1,…,ℱ_r⊂ genfrac[]0ptVk, where genfrac[]0ptVk denotes the set of k-dimensional subspaces of V. Suppose that F_1∩⋯∩ F_r≠{0} holds for all F_i∈ℱ_i, 1≤ i≤ r. Then we show that ∏_i=1^r|ℱ_i|≤genfrac[]0ptn−1k−1, provided n−k is sufficiently large for fixed q and r. Moreover, equality holds if and only if there is a common line L such that every family ℱ_i consists of all k-dimensional subspaces containing the line L. One of the main tools of the proof is a junta theorem concerning intersecting linear maps obtained by Ellis, Kindler, and Lifshitz.
ER -