549.lean`

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 549

References:

erdosproblems.com/549
[Bu74] Burr, S. A., Generalized Ramsey theory for graphs—a survey. Graphs and combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973) (1974), 52-75.
[NSZ16] S. Norin and Y. R. Sun and Y. Zhao, Asymptotics of Ramsey numbers of double stars.

Imports (1)

FormalConjecturesUtil

theorem Erdos549.erdos_549 Documented declaration: erdos_549 sorry line 46 source
Source documentation
If $T$ is a tree which is a bipartite graph with $k$ vertices in one class and $2k$ vertices in the other class then $$R(T)=4k-1.$$
This is false: Norin, Sun, and Zhao [NSZ16] have proved that if $T$ is the union of two stars on $k$ and $2k$ vertices, with an edge joining the centre of the two stars, then $R(T)\geq (4.2-o(1))k$, and conjectured that $R(T)=(4.2+o(1))k$.
This problem is #15 in Ramsey Theory in the graphs problem collection.

depends on sorry what this means
Statement uses: Iff, False, Nat, LE.le, OfNat.ofNat, SimpleGraph, Sum, Fin, HMul.hMul, SimpleGraph.IsTree, Not, SimpleGraph.Adj, Sum.inl, Sum.inr, Eq, SimpleGraph.diagonalGraphRamsey, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_549 : answer(False) ↔
    ∀ (k : ℕ) (hk : 2 ≤ k) (T : SimpleGraph (Fin k ⊕ Fin (2 * k))),
      T.IsTree →
      (∀ x₁ x₂, ¬ T.Adj (Sum.inl x₁) (Sum.inl x₂)) →
      (∀ y₁ y₂, ¬ T.Adj (Sum.inr y₁) (Sum.inr y₂)) →
      SimpleGraph.diagonalGraphRamsey T = 4 * k - 1 := by
…
```
Mentions: SimpleGraph.diagonalGraphRamsey