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PreprintCC BY 4.0arXiv

A non-trivial bound for 3AP-intersecting families

  • Peter Keevash
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Abstract

A family F of subsets of [n] is 3AP-intersecting if every two members have intersection containing a non-trivial three-term arithmetic progression. We prove that there is an absolute constant c>0 such that any such F has size at most (1/2 − c)2n. This is the first non-trivial progress towards a conjecture of Simonovits and Sós that the maximum possible size is 2n−3. More generally, we show the same bound for H-intersecting families whenever H is a $3$-graph on [n] with bounded codegrees. A clique shows that this is sharp, in that the bounded codegree assumption cannot be removed.

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Cite

BibTeXRIS
BibTeX
@misc{keevash2026nontrivial,
  title = {{A non-trivial bound for 3AP-intersecting families}},
  author = {Peter Keevash},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18870},
  eprint = {2609.18870},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18870},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A non-trivial bound for 3AP-intersecting families
AU  - Peter Keevash
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18870
UR  - https://arxiv.org/abs/2609.18870
AB  - A family F of subsets of [n] is 3AP-intersecting if every two members have intersection containing a non-trivial three-term arithmetic progression. We prove that there is an absolute constant c>0 such that any such F has size at most (1/2 − c)2^n. This is the first non-trivial progress towards a conjecture of Simonovits and Sós that the maximum possible size is 2^n−3. More generally, we show the same bound for H-intersecting families whenever H is a $3$-graph on [n] with bounded codegrees. A clique shows that this is sharp, in that the bounded codegree assumption cannot be removed.
ER  -