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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.
Open on the site (last edited 1 April 2026), whose commentary
credits Bucić, Chen and Ma [BCM26] with the affirmative answer for every
and records nothing for . Their Theorem 1.2 (arXiv:2603.18952v1, 19 March
2026, a preprint) gives
over the whole range
, and 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, and 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"). The frontmatter standing is derived from the claim pages: two
accepted full claims, each a Lean proof of the full statement for every
that this corpus built and audited, settle the problem on formalized evidence.
Asad Shahab's claim, filed on the site first (the site's proof claim 358, before
the project's 367, both on 27 September 2026), is a seven-cycle proof with a
Lean development for every , built here at its pinned commit and audited
clause by clause against the Statement (see the Formalization paragraph)
(claim page (Shahab, 2026)). The
project's claim
L17, a
kernel-checked Lean proof of the full statement for every whose
statement the project's fidelity review and grade of 2026-10-02 audited (see the
Formalization paragraph), is the other
(claim page (Plasma AI, 2026)); its
tier-2 standing is the project's own verification, not community acceptance. No
outside review of either proof is known. Bucić, Chen and Ma's theorem is an
accepted partial claim covering on formalized evidence, the
branches of both developments (the site's commentary credits the theorem, but on
a problem the site labels OPEN that credit is not acceptance)
(claim page (Bucić, Chen and Ma, 2026)).