79.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 79

References:

erdosproblems.com/79
[EFRS93] Erdős, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399.
[Wi24] Wigderson, Y., Infinitely many minimally non-Ramsey size linear graphs. arXiv:2409.05931 (2024).

Imports (1)

FormalConjecturesUtil

theorem Erdos79.erdos_79 We say $G$ is Ramsey size linear if $R(G,H)\ll m$ for all graphs $H$ with $m$ edges and no isolated vertices sorry line 47 source
Source documentation
We say $G$ is Ramsey size linear if $R(G,H)\ll m$ for all graphs $H$ with $m$ edges and no isolated vertices.
Are there infinitely many graphs $G$ which are not Ramsey size linear but such that all of its proper subgraphs are?
Asked by Erdős, Faudree, Rousseau, and Schelp [EFRS93]. $K_4$ was long the only known example. Wigderson [Wi24] proved that there are infinitely many such graphs.

depends on sorry what this means
Statement uses: Iff, True, Nat, Exists, LE.le, SimpleGraph, Fin, And, Not, SimpleGraph.IsRamseySizeLinear, SimpleGraph.Subgraph, LT.lt, Top.top, SimpleGraph.Subgraph.verts, Membership.mem, Set, Classical.propDecidable, SimpleGraph.Subgraph.coe
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_79 : answer(True) ↔
    ∀ (N : ℕ), ∃ (n : ℕ) (_ : N ≤ n) (G : SimpleGraph (Fin n)),
      ¬ G.IsRamseySizeLinear ∧
      ∀ H : G.Subgraph, H < ⊤ → H.coe.IsRamseySizeLinear := by
…
```