The sharp constant in random triangle removal at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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The sharp constant in random triangle removal

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	counting, coloring	Levels:	1
Category:	Combinatorics	Lean version:	YES! ✔
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The sharp terminal leave in random triangle removal. Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to $n^{3/2}/(2\sqrt2)$, with mean-square convergence after normalization by n^3/2. This proves the triangle case of the Joos–Kühn sharp-constant conjecture.

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LEVEL 1 The sharp terminal leave in random triangle removal

released 2026-09-25 | 1 theorem · 7 lemmas · 12 proofs · 10,810 words | PLAY LEVEL 1 » (pdf)

Starting from the complete graph on n vertices, repeatedly remove the three edges of a uniformly chosen remaining triangle. We prove that the number of edges left at termination, divided by n^3/2, converges in L² to $1/(2\sqrt2)$. This proves the triangle case of the sharp-constant conjecture of Joos and Kühn. In particular, the same limit holds in probability and for the normalized expectation.

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