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Define the anti-Ramsey number as the smallest such that there is a graph with vertices and edges with an -colouring of its edges in which every copy of has entirely distinct edge colours.
Is it true that, for all ,
Source: erdosproblems.com/809
An accepted solution exists. The statement is true.
Labeled open on erdosproblems.com (last edited 1 April 2026), whose
commentary credits Bucić, Chen and Ma [BCM26] with the affirmative answer for
every and records nothing for . The frontmatter standing is derived
from the claim pages. Two accepted full claims settle the problem on
formalized evidence. Each is a Lean proof of the full statement for every
that this corpus built and audited. No outside review of either proof is
known. One is Asad Shahab's claim, filed on erdosproblems.com first
(erdosproblems.com's proof claim 358, before the project's 367, both on 27
September 2026). It is a seven-cycle proof with a Lean development for every
. That development was built here at its pinned commit and audited clause
by clause against the Statement (see the Formalization paragraph)
(claim page (Shahab, 2026)). The
other is the project's claim
L17, a
kernel-checked Lean proof of the full statement for every , public on
GitHub from 24 September 2026, before Shahab's (see "Outside claims and
priority"). Its statement was audited by the project's fidelity review and grade
of 2026-10-02 (see the Formalization paragraph)
(claim page (Plasma AI, 2026)). Its
tier-2 standing is the project's own verification, not community acceptance.
Bucić, Chen and Ma's theorem is an accepted partial claim covering on
formalized evidence, namely the branches of both developments
(claim page (Bucić, Chen and Ma, 2026)).
The erdosproblems.com commentary credits the theorem, but on a problem Bloom's
catalog labeled OPEN that credit is not acceptance. Bucić, Chen and Ma's Theorem
1.2 (arXiv:2603.18952v1, 19 March 2026, a preprint with no journal record found
on 2026-09-18) gives over
the whole range . At
this is . The case () is
outside that theorem. For it, the 1989 paper gives only the lower bound
and the two-clique coloring that
its authors say would make best possible. No refereed source found
determines the constant. Shahab's preprint arXiv:2609.38286 (29 September
2026), unrefereed and not read here, states it (see "Outside claims and
priority").