feat(ErdosProblems): 15x ramsey theory formalizations#3588
feat(ErdosProblems): 15x ramsey theory formalizations#3588ryantuck wants to merge 6 commits intogoogle-deepmind:mainfrom
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Thanks! Some first remarks:
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@mo271 thanks for the feedback! As advised, I had Opus 4.6 do the following (fine-grain markdown session logs included to show the work): 1 - 3c168fa consolidated reused definitions among my erdos contributions (session logs)
2 - 6b288cd also consolidated wikipedia Ramsey code for reuse (session logs)
3 - 45fd907 holistically assessed whether all Ramsey-related code was in an appropriate place or needed further consolidation, and clarified some references in the erdos file implementations (session logs)
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YaelDillies
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YaelDillies
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Please extend my comments accordingly to the other problems
| ∀ ε : ℝ, ε > 0 → | ||
| ∃ L₀ : ℕ, ∀ l : ℕ, l ≥ L₀ → | ||
| |(graphRamseyNumber k (l + 1) : ℝ) / (graphRamseyNumber k l : ℝ) - 1| ≤ ε := by |
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Can you state this using Filter.Tendsto, Filter.atTop and TopologicalSpace.nhds instead?
| /-- | ||
| Erdős Problem 1014 [Er71, p.99]: | ||
| For fixed $k \geq 3$, | ||
| $$\lim_{l \to \infty} R(k, l+1) / R(k, l) = 1,$$ | ||
| where $R(k, l)$ is the Ramsey number. | ||
| Formulated as: for every $\varepsilon > 0$, there exists $L_0$ such that for all $l \geq L_0$, | ||
| $|R(k, l+1) / R(k, l) - 1| \leq \varepsilon$. | ||
| -/ | ||
| @[category research open, AMS 5] | ||
| theorem erdos_1014 (k : ℕ) (hk : k ≥ 3) : |
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Can you state the k = 3 separately since it already is open?
| *Reference:* [erdosproblems.com/1014](https://www.erdosproblems.com/1014) | ||
| [Er71] Erdős, P., _Topics in combinatorial analysis_, pp. 95-99, 1971. |
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The article seems to only be 19 pages long. Please check all references!
| ∃ K₀ : ℕ, ∀ k : ℕ, k ≥ K₀ → | ||
| (diagRamseyNumber k : ℝ) ≥ C * (k : ℝ) * (2 : ℝ) ^ ((k : ℝ) / 2) := by |
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Again, please use Filter primitives here
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| *Reference:* [erdosproblems.com/1030](https://www.erdosproblems.com/1030) | ||
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| If $R(k,l)$ is the Ramsey number then prove the existence of some $c > 0$ such that |
| See also problems [544](https://www.erdosproblems.com/544) and | ||
| [1014](https://www.erdosproblems.com/1014). | ||
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| OEIS: [A000791](https://oeis.org/A000791), [A059442](https://oeis.org/A059442). |
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What's the relevance of the first one?
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Followup to #3422. Contains 15x formalizations for open problems relating to Ramsey theory.
Problems: