FormalConjectures.ErdosProblems.«564»

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import FormalConjecturesUtil

Erdős Problem 564

Reference: erdosproblems.com/564


namespace Erdos564


open Combinatorics Real Filter

Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices.

Is there some constant $c>0$ such that $$ R_3(n) \geq 2^{2^{cn}}? $$

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_564 : answer(sorry) ↔
    ∃ c > 0, ∀ᶠ n in atTop, (2 : ℝ)^(2 : ℝ)^(c * n) ≤ hypergraphRamsey 3 n := by⊢ True ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, 2 ^ 2 ^ (c * n) ≤ ↑(hypergraphRamsey 3 n)
  sorryAll goals completed! 🐙


end Erdos564