914.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 914

References:

erdosproblems.com/914
[CoHa63] Corrádi, K. and Hajnal, A., On the maximal number of independent circuits in a graph. Acta Math. Acad. Sci. Hungar. (1963), 423-439.
[HaSz70] Hajnal, A. and Szemerédi, E., Proof of a conjecture of P. Erdős. (1970), 601-623.
[KiKo08] Kierstead, H. A. and Kostochka, A. V., A short proof of the Hajnal-Szemerédi theorem on equitable colouring. Combin. Probab. Comput. (2008), 265-270.

Imports (1)

FormalConjecturesUtil

theorem Erdos914.erdos_914 Let $r\geq 2$ and $m\geq 1$ sorry line 46 source
Source documentation
Let $r\geq 2$ and $m\geq 1$. Every graph with $rm$ vertices and minimum degree at least $m(r-1)$ contains $m$ vertex disjoint copies of $K_r$.
When $r=2$ this follows from Dirac's theorem. Corrádi and Hajnal [CoHa63] proved this when $r=3$. Hajnal and Szemerédi [HaSz70] proved this for all $r\geq 4$.
A shorter proof was given by Kierstead and Kostochka [KiKo08].

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, Eq, Fintype.card, HMul.hMul, HSub.hSub, SimpleGraph.minDegree, Exists, Fin, Finset, And, SimpleGraph.IsNClique, Pairwise, Disjoint
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_914 {r m : ℕ} (hr : 2 ≤ r) (hm : 1 ≤ m) {V : Type*} [Fintype V]
    (G : SimpleGraph V) [DecidableRel G.Adj] (hV : Fintype.card V = r * m)
    (hdeg : m * (r - 1) ≤ G.minDegree) :
    ∃ K : Fin m → Finset V, (∀ i, G.IsNClique r (K i)) ∧
      Pairwise fun i j => Disjoint (K i) (K j) := by
…
```
theorem Erdos914.erdos_914.variants.equitable_colouring Documented declaration: equitable_colouring sorry line 59 source
Source documentation
Equivalently, every graph with $rm$ vertices and maximum degree at most $m-1$ has a proper vertex colouring with $m$ colours in which every colour class has exactly $r$ vertices (an equitable colouring).

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, Eq, Fintype.card, HMul.hMul, SimpleGraph.maxDegree, HSub.hSub, Exists, SimpleGraph.Coloring, Fin, Set.ncard, SimpleGraph.Coloring.colorClass
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.
```
theorem erdos_914.variants.equitable_colouring {r m : ℕ} (hr : 2 ≤ r) (hm : 1 ≤ m) {V : Type*}
    [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj] (hV : Fintype.card V = r * m)
    (hdeg : G.maxDegree ≤ m - 1) :
    ∃ C : G.Coloring (Fin m), ∀ i, (C.colorClass i).ncard = r := by
…
```