FormalConjectures.ErdosProblems.«566»

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import FormalConjecturesUtil

Erdős Problem 566

References:

erdosproblems.com/566

[EFRS93] Erdős, Faudree, Rousseau and Schelp, Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399.


namespace Erdos566


open SimpleGraph

Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then $\hat{r}(G,H) \ll m$?

In other words: if $G$ is sparse (every induced subgraph on $k$ vertices has $≤ 2k-3$ edges), is $G$ Ramsey size linear?

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_566 : answer(sorry) ↔
    ∀ (p : ℕ) (G : SimpleGraph (Fin p)),
      -- G is sparse: every induced subgraph on k ≥ 2 vertices has ≤ 2k - 3 edges
      (∀ S : Finset (Fin p), 2 ≤ S.card → (G.induce S).edgeSet.ncard ≤ 2 * S.card - 3) →
      -- Then G is Ramsey size linear
      ∃ c > (0 : ℝ), ∀ (n : ℕ) (H : SimpleGraph (Fin n)) [DecidableRel H.Adj],
        -- H has no isolated vertices
        (∀ v, 0 < H.degree v) →
        -- r̂(G,H) ≤ c · m
        (sizeRamsey G H : ℝ) ≤ c * H.edgeSet.ncard := by⊢ True ↔
  ∀ (p : ℕ) (G : SimpleGraph (Fin p)),
    (∀ (S : Finset (Fin p)), 2 ≤ S.card → (induce (↑S) G).edgeSet.ncard ≤ 2 * S.card - 3) →
      ∃ c > 0,
        ∀ (n : ℕ) (H : SimpleGraph (Fin n)) [inst : DecidableRel H.Adj],
          (∀ (v : Fin n), 0 < H.degree v) → ↑(G.sizeRamsey H) ≤ c * ↑H.edgeSet.ncard
  sorryAll goals completed! 🐙


end Erdos566