Non-trivial intersecting families for direct products

@misc{wang2026nontrivial,
  title = {{Non-trivial intersecting families for direct products}},
  author = {Jian Wang and Biao Wu and Huajun Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29044},
  eprint = {2610.29044},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29044},
}

TY  - UNPB
TI  - Non-trivial intersecting families for direct products
AU  - Jian Wang
AU  - Biao Wu
AU  - Huajun Zhang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29044
UR  - https://arxiv.org/abs/2610.29044
AB  - A family of sets is intersecting if any two of its sets intersect, and non-trivial intersecting if there is not any element which appears in every set of the family. The study of intersection problems (or Erdős-Ko-Rado problems) on families of sets is one of the most important topics in extremal combinatorics. A natural extension of the Erdős-Ko-Rado problem in direct products setting was introduced by Frankl, and developed by Alon and Katona et al. In 2018, Kwan, Sudakov and Vieira [J. Combin. Theory, Ser. A 156 (2018), 44--60] determine the maximum size of a non-trivial intersecting direct product family under the assumption that the sizes of all parts are sufficiently large relative to the set sizes in all parts. Meanwhile, they raised the question of whether this result holds for the size of each part which depends only on its corresponding set size, and the question of characterizing the extremal families. In the present paper, we answere these two questions. Furthermore, the bounds on the size of each part in our result are best possible.1
ER  -