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Combinatorics
Date added
Erdős problem 617
Let r≥3. If the edges of Kr2+1 are r
-coloured then there exist
r+1
vertices with at
least one colour missing on the edges of the induced
Kr+1
.
In other words, there is no balanced colouring.
A conjecture of Erdős and Gyárfás [ErGy99].
$2,666 bounty·1 piece from 1 person·last one 9 days ago
week of 18 Sept 2025 · nothing publishedweek of 25 Sept 2025 · nothing publishedweek of 2 Oct 2025 · nothing publishedweek of 9 Oct 2025 · nothing publishedweek of 16 Oct 2025 · nothing publishedweek of 23 Oct 2025 · nothing published
Solve it
A complete proof or refutation in Lean. The first one the validator accepts takes the whole bounty.
∀ r ≥ 3,
∀ {V : Type} [inst : Fintype V] [DecidableEq V],
Fintype.card V = r ^ 2 + 1 →
∀ (coloring : Sym2 V → Fin r), ∃ S k, S.card = r + 1 ∧ ∀ u ∈ S, ∀ v ∈ S, u ≠ v → coloring s(u, v) ≠ k
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
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