552.lean`

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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 552

References:

erdosproblems.com/552
[BEFRS89] Burr, S. and Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Some complete bipartite graph-tree Ramsey numbers. Graph theory in memory of G. A. Dirac (Sandbjerg, 1985) (1989), 79-89.
[Ch97] Chen, G., A result on $C_4$-star Ramsey numbers. Discrete Mathematics 163 (1997), 243-246.

Imports (1)

FormalConjecturesUtil

theorem Erdos552.erdos_552.parts.i Determine the Ramsey number $$R(C_4, S_n),$$ where $S_n=K_{1,n}$ is the star on $n+1$ vertices sorry line 46 source
Source documentation
Determine the Ramsey number $$R(C_4, S_n),$$ where $S_n=K_{1,n}$ is the star on $n+1$ vertices.
A problem of Burr, Erdős, Faudree, Rousseau, and Schelp [BEFRS89].
This problem is #19 in Ramsey Theory in the graphs problem collection.

depends on sorry what this means
Statement uses: Nat, Eq, SimpleGraph.graphRamsey, Fin, OfNat.ofNat, Sum, SimpleGraph.cycleGraph, completeBipartiteGraph, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_552.parts.i :
    ∀ (n : ℕ),
      SimpleGraph.graphRamsey (SimpleGraph.cycleGraph 4)
        (completeBipartiteGraph (Fin 1) (Fin n)) =
        answer(sorry) := by
…
```
Mentions: SimpleGraph.graphRamsey
theorem Erdos552.erdos_552.parts.ii In particular, is it true that, for any $c > 0$, there are infinitely many $n$ such that $$R(C_4, S_n) \leq n + \sqrt{n} - c?$$ sorry line 58 source
Source documentation
In particular, is it true that, for any $c > 0$, there are infinitely many $n$ such that $$R(C_4, S_n) \leq n + \sqrt{n} - c?$$

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Set.Infinite, Nat, Set.ofPred, LE.le, SimpleGraph.graphRamsey, Fin, Sum, SimpleGraph.cycleGraph, completeBipartiteGraph, HSub.hSub, HAdd.hAdd, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_552.parts.ii : answer(sorry) ↔
    ∀ (c : ℝ), 0 < c →
      Set.Infinite {n : ℕ |
        (SimpleGraph.graphRamsey (SimpleGraph.cycleGraph 4)
          (completeBipartiteGraph (Fin 1) (Fin n)) : ℝ) ≤ (n : ℝ) + Real.sqrt n - c} := by
…
```
Mentions: c, SimpleGraph.graphRamsey
theorem Erdos552.erdos_552.variants.succ_le_add_two sorry line 72 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, SimpleGraph.graphRamsey, Fin, Sum, HAdd.hAdd, SimpleGraph.cycleGraph, completeBipartiteGraph
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.