81.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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 81

References:

erdosproblems.com/81
[EOZ93] Erdős, Paul and Ordman, Edward T. and Zalcstein, Yechezkel, Clique partitions of chordal graphs. Combin. Probab. Comput. (1993), 409–415.
[CEO94] Chen, Guan-Tao and Erdős, Paul and Ordman, Edward T., Clique partitions of split graphs. Combinatorics, graph theory, algorithms and applications (Beijing, 1993) (1994), 21–30.

Imports (1)

FormalConjecturesUtil

theorem Erdos81.erdos_81 Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$ sorry line 41 source
Source documentation
Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $n^2/6+O(n)$ many cliques?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, Nat, SimpleGraph, Fin, SimpleGraph.IsChordal, Finset, And, SimpleGraph.IsCliqueEdgePartition, LE.le, Finset.card, HAdd.hAdd, HDiv.hDiv, HPow.hPow, OfNat.ofNat, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_81 :
    answer(sorry) ↔ ∃ C : ℝ, ∀ n : ℕ, ∀ G : SimpleGraph (Fin n), G.IsChordal →
      ∃ P : Finset (Finset (Fin n)), G.IsCliqueEdgePartition P ∧
        (P.card : ℝ) ≤ (n : ℝ) ^ 2 / 6 + C * n := by
…
```
theorem Erdos81.erdos_81.variants.split Chen, Erdős, and Ordman [CEO94] have shown that any split graph can be partitioned into $\frac{3}{16}n^2+O(n)$ many cliques sorry line 50 source
Source documentation
Chen, Erdős, and Ordman [CEO94] have shown that any split graph can be partitioned into $\frac{3}{16}n^2+O(n)$ many cliques.

depends on sorry what this means
Statement uses: Exists, Real, Nat, SimpleGraph, Fin, SimpleGraph.IsSplitGraph, Finset, And, SimpleGraph.IsCliqueEdgePartition, LE.le, Finset.card, HAdd.hAdd, HDiv.hDiv, HMul.hMul, OfNat.ofNat, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_81.variants.split :
    ∃ C : ℝ, ∀ n : ℕ, ∀ G : SimpleGraph (Fin n), G.IsSplitGraph →
      ∃ P : Finset (Finset (Fin n)), G.IsCliqueEdgePartition P ∧
        (P.card : ℝ) ≤ 3 * (n : ℝ) ^ 2 / 16 + C * n := by
…
```