{"ID":22964819,"CreatedAt":"2026-09-17T02:29:35.102872001Z","UpdatedAt":"2026-09-17T02:29:35.102872001Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18870","arxiv_id":"2609.18870","title":"A non-trivial bound for 3AP-intersecting families","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 $c\u003e0$ such that any such $F$ has size at most $(\\tfrac12 - c)2^n$. 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.","short_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 $c\u003e0$ such that any such $F$ has size at most $(\\tfrac12 - c)2^n$. This is the first non-trivial progress towards a conjectu...","url_abs":"https://arxiv.org/abs/2609.18870","url_pdf":"https://arxiv.org/pdf/2609.18870v1","authors":"[\"Peter Keevash\"]","published":"2026-09-16T16:08:10Z","proceeding":"math.CO","tasks":"[\"math.CO\"]","methods":"[]","has_code":false}