568.lean`

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 568

References:

erdosproblems.com/568
[EFRS93] Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399.

Imports (1)

FormalConjecturesUtil

theorem Erdos568.erdos_568 Let $G$ be a graph such that $R(G,T_n)\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\ll n^2$ sorry line 43 source
Source documentation
Let $G$ be a graph such that $R(G,T_n)\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices, $$R(G,H)\ll m?$$
In other words, is $G$ Ramsey size linear?
This problem is #33 in Ramsey Theory in the graphs problem collection.

depends on sorry what this means
Statement uses: Iff, True, Fintype, SimpleGraph, Exists, Real, And, GT.gt, OfNat.ofNat, Nat, Fin, SimpleGraph.IsTree, LE.le, SimpleGraph.graphRamsey, HMul.hMul, SimpleGraph.completeGraph, HPow.hPow, SimpleGraph.IsRamseySizeLinear
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_568 : answer(sorry) ↔
    ∀ (V : Type) [Fintype V] (G : SimpleGraph V),
      (∃ c₁ > (0 : ℝ), ∀ (n : ℕ) (T : SimpleGraph (Fin n)),
        T.IsTree → (SimpleGraph.graphRamsey G T : ℝ) ≤ c₁ * n) →
      (∃ c₂ > (0 : ℝ), ∀ (n : ℕ),
        (SimpleGraph.graphRamsey G (SimpleGraph.completeGraph (Fin n)) : ℝ) ≤ c₂ * (n : ℝ) ^ 2) →
      G.IsRamseySizeLinear := by
…
```
Mentions: SimpleGraph.graphRamsey