The maximum t-diversity of t-intersecting families of permutations

@misc{cao2026maximum,
  title = {{The maximum $t$-diversity of $t$-intersecting families of permutations}},
  author = {Yuhang Cao and Gennian Ge and Jian Wang and Jialuo Wang and Xiaochen Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12391},
  eprint = {2609.12391},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12391},
}

TY  - UNPB
TI  - The maximum t-diversity of t-intersecting families of permutations
AU  - Yuhang Cao
AU  - Gennian Ge
AU  - Jian Wang
AU  - Jialuo Wang
AU  - Xiaochen Zhao
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12391
UR  - https://arxiv.org/abs/2609.12391
AB  - The study of t-intersecting families in symmetric groups received lots of attention in the last two decades. In this paper, we study two different kinds of stability results for t-intersecting families in symmetric groups. Let Σ_n denote the symmetric group on {1,2,…,n}. The t-diversity γ_t(ℱ) of ℱ⊆Σ_n is defined as the minimum number of members of ℱ whose deletion results in a family with transversal number t. The star t-diversity γ_t^⋆(ℱ) is defined as the minimum number of members of ℱ whose deletion results in a t-star. For n relatively large with respect to t, we determine the best possible bounds for both γ_t(ℱ) and γ_t^⋆(ℱ) over all t-intersecting families ℱ⊆Σ_n. The equality holding conditions are characterized. For γ_t(ℱ), the extremal case is generated by all 2t-subsets of a 3t-partial permutation or, when t=2, the family complements of the lines of the Fano plane. For γ_t^⋆(ℱ), the extremal family is generated by all (t+1)-subsets of a (t+2)-partial permutation.
ER  -