551.lean`

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 551

References:

erdosproblems.com/551
[BoEr73] Bondy, J. A. and Erdős, P., Ramsey numbers for cycles in graphs. J. Combin. Theory Ser. B (1973), 46-54.
[Ni05] Nikiforov, V., The cycle-complete graph Ramsey numbers. Combin. Probab. Comput. (2005), 349-370.
[KLS21] Keevash, P., Long, E. and Skokan, J., Cycle-complete Ramsey numbers. Int. Math. Res. Not. IMRN (2021), 277-302.

Imports (1)

FormalConjecturesUtil

theorem Erdos551.erdos_551 Prove that $$R(C_k,K_n)=(k-1)(n-1)+1$$ for $k\geq n\geq 3$ (except when $n=k=3$) sorry line 48 source
Source documentation
Prove that $$R(C_k,K_n)=(k-1)(n-1)+1$$ for $k\geq n\geq 3$ (except when $n=k=3$).
Asked by Erdős, Faudree, Rousseau, and Schelp. This problem is #18 in Ramsey Theory in the graphs problem collection.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Not, And, Eq, SimpleGraph.graphRamsey, Fin, SimpleGraph.cycleGraph, SimpleGraph.completeGraph, HAdd.hAdd, HMul.hMul, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_551 :
    ∀ (k n : ℕ), 3 ≤ n → n ≤ k → ¬(n = 3 ∧ k = 3) →
      SimpleGraph.graphRamsey (SimpleGraph.cycleGraph k)
        (SimpleGraph.completeGraph (Fin n)) = (k - 1) * (n - 1) + 1 := by
…
```
Mentions: SimpleGraph.graphRamsey
theorem Erdos551.erdos_551.variants.sufficiently_large For sufficiently large $n$ and every $k\geq n$, $R(C_k,K_n)=(k-1)(n-1)+1$ sorry line 62 source
Source documentation
For sufficiently large $n$ and every $k\geq n$, $R(C_k,K_n)=(k-1)(n-1)+1$.
Keevash, Long, and Skokan [KLS21] have proved this identity when $k\geq C\frac{\log n}{\log\log n}$ for some constant $C$, thus establishing the conjecture for sufficiently large $n$.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, LE.le, Eq, SimpleGraph.graphRamsey, Fin, SimpleGraph.cycleGraph, SimpleGraph.completeGraph, HAdd.hAdd, HMul.hMul, HSub.hSub, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_551.variants.sufficiently_large :
    ∀ᶠ n : ℕ in atTop, ∀ k : ℕ, n ≤ k →
      SimpleGraph.graphRamsey (SimpleGraph.cycleGraph k)
        (SimpleGraph.completeGraph (Fin n)) = (k - 1) * (n - 1) + 1 := by
…
```
Mentions: SimpleGraph.graphRamsey