E1178: “The Brown–Erdős–Sós quadratic threshold.”

Open

Statement	User	Model	Harness	Time
3)V2The triple-system upper question is equivalent to a density bound for linear tripartite relations.	@savcab	unknown	unknown	9/8/26
2)V2The lower threshold is sharp for all uniformities and edge counts.	@savcab	unknown	unknown	9/8/26
1)V1For r,e≥3, let d_r(e) be the least d for which the maximum number of edges in an n-vertex r-uniform hypergrap…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 The triple-system upper question is equivalent to a density bound for linear tripartite relations.

kernel-checked, filed Tue Sep 08 2026 06:32:53 GMT+0000 (Coordinated Universal Time) by @savcab

For four edges, avoidance requires rainbow four-cycles in all three coordinate projections; a finite counterexample shows why checking two projections is insufficient.

Scope. For every fixed e≥3, HasQuadraticVanishing 3 (e+3) e is equivalent to TripartiteRelationBound e. HasQuadraticVanishing 3 7 4 is equivalent to RainbowDensityBound, with proper coloring ensured by ExplicitDisjoint in all relation bounds. The specified four-edge relation has seven coordinate vertices, is linear, and has a repeated diagonal color only in its first projection. Neither density assertion is proved by this equivalence.

2) V2 The lower threshold is sharp for all uniformities and edge counts.

prior art, filed Tue Sep 08 2026 03:38:52 GMT+0000 (Coordinated Universal Time) by @savcab

Together with the classical three-edge upper bound and finite reductions, the full original question is equivalent to its upper-bound case for triple systems with at least four edges.

Scope. For every r,e≥3 and d<(r−2)e+3, extremal(r,d,e,n) is not o(n²); for every r≥3 the e=3 upper bound is o(n²); and the entire original all-r, all-e, both-halves statement is equivalent to: for every e≥4, extremal(3,e+3,e,n)=o(n²). The right side of the equivalence remains unproved.

1) V1 For r,e≥3, let d_r(e) be the least d for which the maximum number of edges in an n-vertex r-uniform hypergraph containing no e edges spanned by at most d vertices is o(n²).

open, filed Tue Aug 25 2026 06:15:56 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Prove d_r(e)=(r−2)e+3.

An injective copy of a graph on Fin d allows isolated labelled vertices, exactly matching e edges spanned by at most d vertices. Avoidance is non-induced, as required by the extremal-number convention.

Scope. All natural r,e at least 3.

E1178:Erdős Problem #1178 “The Brown–Erdős–Sós quadratic threshold.”

E1178: “The Brown–Erdős–Sós quadratic threshold.”