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module
public import FormalConjecturesUtilErdős Problem 87
References:
[Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.
@[expose] public sectionopen Filternamespace Erdos87Let $0 < \epsilon < 1$. Is it true that, if $k$ is sufficiently large, then $$R(G) > (1-\epsilon)^k R(k)$$ for every graph $G$ with chromatic number $\chi(G)=k$?
The restriction $\epsilon < 1$ excludes negative bases in $(1-\epsilon)^k$.
This problem is #12 in Ramsey Theory in the graphs problem collection.
@[category research open, AMS 5]
theorem erdos_87.parts.i : answer(sorry) ↔
∀ ε > (0 : ℝ), ε < 1 → ∀ᶠ k : ℕ in atTop,
∀ (V : Type) [Fintype V] (G : SimpleGraph V), G.chromaticNumber = (k : ℕ∞) →
(SimpleGraph.diagonalGraphRamsey G : ℝ) >
(1 - ε) ^ k * (SimpleGraph.diagonalRamsey k : ℝ) := ⊢ True ↔
∀ ε > 0,
ε < 1 →
∀ᶠ (k : ℕ) in atTop,
∀ (V : Type) [inst : Fintype V] (G : SimpleGraph V),
G.chromaticNumber = ↑k → ↑G.diagonalGraphRamsey > (1 - ε) ^ k * ↑(SimpleGraph.diagonalRamsey k)
All goals completed! 🐙Even stronger, is there some $c > 0$ such that, for all large $k$, $$R(G) > c R(k)$$ for every graph $G$ with chromatic number $\chi(G)=k$?
This problem is #13 in Ramsey Theory in the graphs problem collection.
@[category research open, AMS 5]
theorem erdos_87.parts.ii : answer(sorry) ↔
∃ c > (0 : ℝ), ∀ᶠ k : ℕ in atTop,
∀ (V : Type) [Fintype V] (G : SimpleGraph V), G.chromaticNumber = (k : ℕ∞) →
(SimpleGraph.diagonalGraphRamsey G : ℝ) > c * (SimpleGraph.diagonalRamsey k : ℝ) := ⊢ True ↔
∃ c > 0,
∀ᶠ (k : ℕ) in atTop,
∀ (V : Type) [inst : Fintype V] (G : SimpleGraph V),
G.chromaticNumber = ↑k → ↑G.diagonalGraphRamsey > c * ↑(SimpleGraph.diagonalRamsey k)
All goals completed! 🐙-- TODO: Add variants of the problem.
end Erdos87