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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.02698 |
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| _version_ | 1866911644503769088 |
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| author | Ihringer, Ferdinand Kupavskii, Andrey |
| author_facet | Ihringer, Ferdinand Kupavskii, Andrey |
| contents | We study $t$-intersecting and $t$-cross-intersecting families of $k$-dimensional subspaces in finite vector spaces of dimension $n$. We show that all large $t$-intersecting families admit a governing low-dimensional structure for $n \ge 2k+1$. This result, together with its cross-intersecting variant, allows us to prove analogues of several classical extremal set-theoretic results. In particular, we determine the intersecting families with the largest diversity, and we establish a Frankl-type degree-diversity result that generalizes the Hilton-Milner theorem. Our proofs rely on simplification procedures for $t$-intersecting and $t$-cross-intersecting families of subspaces. These procedures are based on the concept of subspace spreadness, a generalization of the classical notion of spreadness for set systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02698 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structure of $t$-Intersecting Families of Vector Spaces Ihringer, Ferdinand Kupavskii, Andrey Combinatorics We study $t$-intersecting and $t$-cross-intersecting families of $k$-dimensional subspaces in finite vector spaces of dimension $n$. We show that all large $t$-intersecting families admit a governing low-dimensional structure for $n \ge 2k+1$. This result, together with its cross-intersecting variant, allows us to prove analogues of several classical extremal set-theoretic results. In particular, we determine the intersecting families with the largest diversity, and we establish a Frankl-type degree-diversity result that generalizes the Hilton-Milner theorem. Our proofs rely on simplification procedures for $t$-intersecting and $t$-cross-intersecting families of subspaces. These procedures are based on the concept of subspace spreadness, a generalization of the classical notion of spreadness for set systems. |
| title | Structure of $t$-Intersecting Families of Vector Spaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.02698 |