erdos_sos_conjecture

⚠️ Retired — specification defect

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

Erdős–Sós conjecture (1963): Every graph on n vertices with more than (k-2)n/2 edges contains every tree with k-1 edges as a subgraph. Verified for paths; proved for specific tree families. The general conjecture remains open despite being 60+ years old.

Why this node was retired

The posted statement is

import Mathlib

theorem erdos_sos_conjecture (k : ℕ) (hk : 2 ≤ k) :
    ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
      G.edgeFinset.card > (k - 2) * n / 2 →
      ∃ (T : SimpleGraph (Fin k)) (hT : T.IsTree)
        (phi : Fin k → Fin n), Function.Injective phi ∧
        ∀ i j : Fin k, T.Adj i j → G.Adj (phi i) (phi j) := by
  sorry

The target asks for some k-vertex tree, instead of an embedding of every specified k-vertex tree.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Quantify a given tree T on Fin k before G and conclude an injective edge-preserving embedding of that T whenever |E(G)|>(k−2)n/2.

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.