FormalConjectures.ErdosProblems.«181»

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module

public import FormalConjecturesUtil

Erdős Problem 181

References:

erdosproblems.com/181

[Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.

[Ti22] Tikhomirov, K., A remark on the Ramsey number of the hypercube. arXiv:2208.14568 (2022).

@[expose] public sectionnamespace Erdos181open SimpleGraph

Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that $$R(Q_n) \ll 2^n.$$

@[category research open, AMS 5]
theorem erdos_181 :
    ∃ C > (0 : ℝ), ∀ n : ℕ,
      (diagonalGraphRamsey (hypercube n) : ℝ) ≤ C * 2 ^ n := by⊢ ∃ C > 0, ∀ (n : ℕ), ↑(hypercube n).diagonalGraphRamsey ≤ C * 2 ^ n
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos181