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Erdős Problem 78: erdos_78

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Added 1 Nov 2026 · not yet checked against the literature · importance 1.50

What the prompt says

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Statement

Let $R(k)$ be the Ramsey number for $K_k$ . Give a constructive proof that $R(k) > C^k$ for some constant $C > 1$ .

Equivalently, give an explicit construction of graphs on $n$ vertices which contain no clique and no independent set of size $\geq c \log n$ , for some constant $c > 0$ .

We formalise "explicit" as: the adjacency relation of the graph on $n$ vertices is decided by a single algorithm that runs in time polynomial in $n$ (see explicitGraph). This is the weakest reasonable notion ("weakly explicit"); even this is open. The strongly explicit version, with time polynomial in $\log n$ , is erdos_78.variants.strongly_explicit.

This problem is #4 in Ramsey Theory in the graphs problem collection.

Lean statement

theorem erdos_78 :
    ∃ c > (0 : ℝ), ∃ adj : List Bool × ℕ × ℕ → Bool, IsPolyTime adj ∧
      ∀ᶠ n in atTop, NoHomogeneousSet (explicitGraph adj n) ⌈c * log n⌉₊ := by
  sorry

Sources(1)

formal-conjectures: FormalConjectures/ErdosProblems/78.lean (opens in a new tab)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/ErdosProblems/78.lean.

Literature events(0)

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Map of approaches(0)

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Attempts(0)

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Papers on this problem(0)

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