546.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 546

References:

erdosproblems.com/546
[Su11] Sudakov, B., A conjecture of Erdős on graph Ramsey numbers. Adv. Math. (2011), 3148-3155.
[AKS03] Alon, N., Krivelevich, M. and Sudakov, B., Turán numbers of bipartite graphs and Ramsey graphs of bounded degree. Combin. Probab. Comput. (2003), 477-483.

Imports (1)

FormalConjecturesUtil

theorem Erdos546.erdos_546 Let $G$ be a graph with no isolated vertices and $m$ edges sorry line 43 source
Source documentation
Let $G$ be a graph with no isolated vertices and $m$ edges. Is it true that $$R(G) \leq 2^{O(m^{1/2})}?$$
This is true, and was proved by Sudakov [Su11].
This problem is #11 in Ramsey Theory in the graphs problem collection.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Nat, Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, LT.lt, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, Eq, Set.ncard, Sym2 and 6 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_546 : answer(True) ↔
    ∃ C > (0 : ℝ), ∀ (m : ℕ) (V : Type) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj],
      (∀ v, 0 < G.degree v) →
      G.edgeSet.ncard = m →
      (SimpleGraph.diagonalGraphRamsey G : ℝ) ≤ 2 ^ (C * Real.sqrt m) := by
…
```
Mentions: SimpleGraph.diagonalGraphRamsey