Ramseys_Theorem

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

For any r, s >= 2 there exists N such that any 2-coloring of edges of K_N contains a red K_r or blue K_s.

Why this node was retired

The posted statement is

theorem Ramseys_Theorem (r s : ℕ) (hr : r ≥ 2) (hs : s ≥ 2) : ∃ N : ℕ, ∀ (coloring : Fin N → Fin N → Bool), (∃ S : Finset (Fin N), S.card = r ∧ ∀ i j, i ∈ S → j ∈ S → i ≠ j → coloring i j = true) ∨ (∃ S : Finset (Fin N), S.card = s ∧ ∀ i j, i ∈ S → j ∈ S → i ≠ j → coloring i j = false) := by sorry

The coloring of ordered pairs need not be symmetric, unlike an edge coloring of an undirected complete graph.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Require coloring(i,j)=coloring(j,i), or color unordered2-element subsets instead.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.