Sharp terminal edge count in random triangle removal
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
For the process that repeatedly removes the three edges of a uniformly selected remaining triangle from $K_n$ until triangle-free, does the final edge count $F_n$ satisfy $E[(F_n/n^{3/2}-1/(2$ $\operatorname{sqrt}(2)))^2]->0$?Short English statement
For the process that repeatedly removes the three edges of a uniformly selected remaining triangle from $K_n$ until triangle-free, does the final edge count $F_n$ satisfy $E[(F_n/n^{3/2}-1/(2$ $\operatorname{sqrt}(2)))^2]->0$?Sources
- S1 Original question source or mathematical context. https://arxiv.org/abs/1203.4223v3.
- S2 Original question source or mathematical context. https://arxiv.org/abs/1203.4223v3.
- S3 Original question source or mathematical context. https://arxiv.org/abs/1203.4223v3.
- S4 Original question source or mathematical context. https://arxiv.org/abs/2412.15039v2.
- S5 Original question source or mathematical context. https://arxiv.org/abs/2412.15039v2.
- S6 Original question source or mathematical context. https://arxiv.org/abs/2412.15039v2.
- S7 Original question source or mathematical context. https://cs.nyu.edu/~spencer/papers/asympacking.pdf.
- S8 Original question source or mathematical context. https://cs.nyu.edu/~spencer/papers/asympacking.pdf.
- S9 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2625
Random triangle-removal terminal constant: The manuscript’s result is exactly the terminal-edge-count mean-square limit E[(F_n/n^(3/2)-1/(2sqrt(2)))^2]->0 for uniform triangle selection until no triangles remain. It concerns the terminal leave rather than an intermediate-time spread estimate.
With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.
Sharp terminal edge count in random triangle removal
OpenAI's claim
Random triangle-removal terminal constant: The final edge count in uniform random triangle removal from $K_n$, divided by $n^{3/2}$, is claimed to approach $1/(2\sqrt2)$ in mean square. Convergence of expectation and in probability follow.Mathematical statement of the claimed result
Mathematical claim 1.
Start from $K_n$ and repeatedly choose uniformly among the remaining triangles, deleting its three edges, until no triangle remains. If $F_n$ is the final edge count, the claim is \[\mathbb E\left[\left(F_n/n^{3/2}-1/(2\sqrt2)\right)^2\right]\longrightarrow0.\] Thus $F_n/n^{3/2}\to1/(2\sqrt2)$ in probability and $\mathbb EF_n/n^{3/2}\to1/(2\sqrt2)$ as well.Outline of the argument
Random triangle-removal terminal constant: The process is stopped while subgraph counts remain controlled, then its continuation is represented by independent triangle priorities. Backward recursive survival queries are unfolded over the growing terminal time interval. Stability and paired-query estimates have errors smaller than the two-root survival scale, permitting first- and second-moment computations with the sharp constant.Formal verification
Family 188 describes Lean coverage for the terminal triangle-removal mean-square limit with its sharp coefficient.Random triangle-removal terminal constant: The manuscript links TriangleRemoval.lean.
The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.
This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.
Status caveat
OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.Sources
- OpenAI, The sharp terminal leave in random triangle removal, 2026-09-25; citation key
OAI:The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - Family 188: Lean scope.
- Comparator TriangleRemoval.json.
- Comparator TriangleRemoval.lean.
- OpenAI catalogue entry.