1022.lean`

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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 1022

References:

erdosproblems.com/1022
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.
[Lo68] Lovász, L., On covering of graphs. Theory of Graphs (Proc. Colloq., Tihany, 1966) (1968), 231-236.
[Wo13b] Wood, D. R., Hypergraph colouring and degeneracy. arXiv:1310.2972 (2013).

Imports (1)

FormalConjecturesUtil

def Erdos1022.SparseImpliesPropertyB Documented declaration: SparseImpliesPropertyB line 39 source
Source documentation
SparseImpliesPropertyB t c asserts that every finite family F of finite sets, all of size at least t, such that for every nonempty finite set X there are < c * |X| many A ∈ F with A ⊆ X, has property B.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Finset, Membership.mem, LE.le, Finset.card, Finset.Nonempty, LT.lt, Finset.hypergraphInduce, HMul.hMul, Finset.HasPropertyB
9 instances, coercions and auxiliary constants left out.
```
def SparseImpliesPropertyB (t : ℕ) (c : ℝ) : Prop :=
…
```
Mentions: c
theorem Erdos1022.erdos_1022 sorry line 59 source
depends on sorry what this means
Statement uses: Iff, False, Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, Erdos1022.SparseImpliesPropertyB
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1022.erdos_1022.variants.lt_two Documented declaration: lt_two sorry line 71 source
Source documentation
This is false, and $c_t<2$ for all $t$: a counterexample is provided by Wood [Wo13b], who constructs, for any $r\geq 2$, a triangle-free $2$-degenerate $r$-uniform hypergraph with chromatic number $3$. A similar counterexample was found independently by KoishiChan in the comments.

depends on sorry what this means
Statement uses: Iff, True, Nat, Real, Erdos1022.SparseImpliesPropertyB, LT.lt, OfNat.ofNat
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_1022.variants.lt_two : answer(True) ↔
    ∀ (t : ℕ) (c : ℝ), SparseImpliesPropertyB t c → c < 2 := by
…
```
Mentions: c, Erdos1022.SparseImpliesPropertyB
theorem Erdos1022.erdos_1022.variants.lovasz Erdős originally conjectured, in this language, that $c_2=1$, which he reports in [Er71] was proved by Lovász sorry line 80 source
Source documentation
Erdős originally conjectured, in this language, that $c_2=1$, which he reports in [Er71] was proved by Lovász.

depends on sorry what this means
Statement uses: IsGreatest, Real, Set.ofPred, Erdos1022.SparseImpliesPropertyB, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1022.variants.lovasz :
    IsGreatest {c : ℝ | SparseImpliesPropertyB 2 c} 1 := by
…
```
Mentions: c, Erdos1022.SparseImpliesPropertyB