Result 188, Combinatorics

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 n3/2/(22)n^{3/2}/(2\sqrt2), with mean-square convergence after normalization by n3/2. This proves the triangle case of the Joos–Kühn sharp-constant conjecture.

Lean formalization Proof

The bigger picture

Why it matters

Repeatedly deleting random triangles from a fully connected network eventually leaves edges that form no triangle. The result identifies the precise leading size of this leftover, showing how predictably a random process can stop short of using every edge.

What changes?

Start with the complete graph on n vertices, with an edge joining every pair. At each step, choose uniformly among the remaining triangles and delete its three edges, stopping when none remain. The manuscript reports that, as n grows, the leftover edge count divided by n to the power three halves converges in mean square to one divided by twice the square root of two. This establishes the triangle case of the Joos–Kühn sharp-constant conjecture.

What does that help mathematicians do?

Mean-square convergence controls both the average outcome and fluctuations: the normalized expectation approaches the stated constant, and the standard deviation becomes negligible compared with n to the power three halves. Thus, a researcher can deduce that deviations by any fixed fraction of this scale become unlikely. The conclusion supplies a precise leading constant, not merely an order of growth, without specifying a convergence rate.

Are there practical applications?

The immediate value is foundational for studying random constructions in combinatorics. The deleted triangles form a collection in which no edge is used twice, so the result quantifies how many edges this particular random procedure leaves uncovered. It describes the coverage achieved by this process, not a guarantee that the resulting collection is optimal.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The sharp terminal leave in random triangle removal

September 25, 2026 25 pages

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 n3/2, converges in L2 to 1/(22)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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026,
  author = {{OpenAI}},
  title = {{The sharp terminal leave in random triangle removal}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026.pdf}{OAI:The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/188.md.

The sharp terminal leave in random triangle removal

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Start with the complete graph on nn vertices and repeatedly delete the three edges of a uniformly chosen remaining triangle. The formalization proves that the number of edges at termination, divided by n3/2n^{3/2}, converges in L2L^2 to 1/(22)1/(2\sqrt2). It also states convergence in probability and convergence of the normalized expectation to the same constant. This is the triangle case of the sharp terminal-leave conjecture of Joos and Kühn.

Comparator links

Result Comparator statement
Sharp terminal edge count in random triangle removal TriangleRemoval.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.