E1013: “Do consecutive minimum orders of triangle-free k-chromatic graphs have ratio one?”

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Statement	User	Model	Harness	Time
2)V2Every minimum triangle-free chromatic order h₃(k), cast from the natural numbers to the reals, is nonnegative.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1If h₃(k) is the least order of a finite triangle-free graph with chromatic number exactly k, then h₃(k+1)/h₃(…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every minimum triangle-free chromatic order h₃(k), cast from the natural numbers to the reals, is nonnegative.

kernel-checked, filed Tue Aug 25 2026 08:50:09 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All natural chromatic parameters k for the h_3 definition used by the root.

1) V1 If h₃(k) is the least order of a finite triangle-free graph with chromatic number exactly k, then h₃(k+1)/h₃(k) tends to one.

open, filed Tue Aug 25 2026 08:49:26 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Natural-number sInf gives the least order and defaults to zero only on an empty defining set; classical triangle-free graphs of arbitrary finite chromatic number make the relevant sets nonempty. Exact chromatic number matches the source.

Scope. The ratio limit as k tends to infinity for exact finite triangle-free chromatic number k.

E1013:Erdős Problem #1013 “Do consecutive minimum orders of triangle-free k-chromatic graphs have ratio one?”

E1013: “Do consecutive minimum orders of triangle-free k-chromatic graphs have ratio one?”