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The complete nontrivial-intersection theorem for vector spaces

  • Dehai Liu
  • Benjian Lv
  • Kaishun Wang
  • Jie Wen
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Abstract

A family of k-subspaces of an n-dimensional vector space over a finite field is t-intersecting if any two members intersect in a subspace of dimension at least t. A t-intersecting family is nontrivial if no t-subspace is contained in all its members. In this paper, we determine the largest nontrivial t-intersecting families for k≥ t+2 and n≥ 2k+1. As a result, the complete nontrivial-intersection theorem for vector spaces is established.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026complete,
  title = {{The complete nontrivial-intersection theorem for vector spaces}},
  author = {Dehai Liu and Benjian Lv and Kaishun Wang and Jie Wen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04270},
  eprint = {2610.04270},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04270},
}
RIS
TY  - UNPB
TI  - The complete nontrivial-intersection theorem for vector spaces
AU  - Dehai Liu
AU  - Benjian Lv
AU  - Kaishun Wang
AU  - Jie Wen
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04270
UR  - https://arxiv.org/abs/2610.04270
AB  - A family of k-subspaces of an n-dimensional vector space over a finite field is t-intersecting if any two members intersect in a subspace of dimension at least t. A t-intersecting family is nontrivial if no t-subspace is contained in all its members. In this paper, we determine the largest nontrivial t-intersecting families for k≥ t+2 and n≥ 2k+1. As a result, the complete nontrivial-intersection theorem for vector spaces is established.
ER  -