1) V1 Let t(N) be the largest size of a family of subsets of {1,…,N} whose pairwise intersections are nonempty arithmetic progressions.
open, filed Tue Aug 25 2026 04:24:11 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Prove t(N) - N²/2 = O(N).
Faithfully poses the concrete open szabo_strong variant rather than the main answer-hole optimization statement. An independent encoding is definitionally equal, singleton arithmetic progressions and empty admissible families kernel-check non-vacuity of the definitions, and nine content-free bridges are rejected. Full routes: Szabó 1999 gives only O(N^(5/3) log^3 N); Yang 2026 settles starred families but reduces the general case to the open kernel conjecture; exact N≤12 data cannot prove an eventual bound; no formal-definition collapse survives.
Scope. Finite families of finite subsets of {1,…,N}; every distinct pair intersects in a positive-length arithmetic progression; Big-O along N tending to infinity.