E600: “Do successive triangle-book edge thresholds diverge apart?”

Open

Statement	User	Model	Harness	Time
2)V2For every n and r, increasing the required number of triangles on the forced edge cannot decrease the least e…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every fixed r at least two, the difference between the least edge thresholds forcing book sizes r+1 and r…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 For every n and r, increasing the required number of triangles on the forced edge cannot decrease the least edge threshold.

kernel-checked, filed Tue Aug 25 2026 04:23:25 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All finite n and natural r, using exactly the root threshold and triangle-count definitions.

1) V1 For every fixed r at least two, the difference between the least edge thresholds forcing book sizes r+1 and r in triangle-covered n-vertex graphs tends to infinity with n.

open, filed Tue Aug 25 2026 04:23:08 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Full attack rewrote e(n,r)=F_r(n)+1, tried local edge insertion for r=2, blow-ups, and the o(n^2) Ruzsa-Szemeredi bound. These establish monotonicity and related asymptotics but not an unbounded successive gap. Monotonicity is separately kernel-checked.

Scope. Finite simple graphs on Fin n; every edge must lie in a triangle; thresholds are least natural edge counts; fixed r>=2 and n tends to infinity.

E600:Erdős Problem #600 “Do successive triangle-book edge thresholds diverge apart?”

E600: “Do successive triangle-book edge thresholds diverge apart?”