21.lean`

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L4DEL: Lean4 Domain Explanation Language

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

6 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 21

References:

erdosproblems.com/21
[Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.
[Er92b] Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.
[Er97f] Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.
[ErLo75] Erdős, P. and Lovász, L., Problems and results on $3$-chromatic hypergraphs and some related questions. (1975), 609--627.
[Ka92b] Kahn, Jeff, On a problem of Erdős and Lovász: random lines in a projective plane. Combinatorica (1992), 417-423.
[Ka94] Kahn, Jeff, On a problem of Erdős and Lovász. II. $n(r)=O(r)$. J. Amer. Math. Soc. (1994), 125-143.
[Tr14] A. Tripathi, A result on intersecting families with maximum transversal size. arXiv:1409.4610 (2014).
[BaWa21] J. Barát and I. M. Wanless, Intersecting and 2-intersecting hypergraphs with maximal covering number: the Erdős-Lovász theme revisited. J. Combin. Des. (2021), 260-286.

Imports (1)

FormalConjecturesUtil

def Erdos21.IsErdosLovaszFamily An Erdős–Lovász family of order n: an intersecting family of n-sets such that every set of size at most n - 1 is disjoint from some member line 53 source
Source documentation
An Erdős–Lovász family of order n: an intersecting family of n-sets such that every set of size at most n - 1 is disjoint from some member.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: And, Membership.mem, Eq, Finset.card, Finset.Nonempty, Inter.inter, LE.le, HSub.hSub, OfNat.ofNat, Exists, Disjoint
10 instances, coercions and auxiliary constants left out.
```
def IsErdosLovaszFamily (n : ℕ) (F : Finset (Finset ℕ)) : Prop :=
…
```
def Erdos21.f f n is the minimal size of an Erdős–Lovász family of order n line 58 source
Source documentation
f n is the minimal size of an Erdős–Lovász family of order n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, Finset, And, Erdos21.IsErdosLovaszFamily, Eq, Finset.card
1 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℕ :=
…
```
theorem Erdos21.erdos_21 sorry line 85 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Filter.Eventually, LE.le, Erdos21.f, HMul.hMul, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos21.erdos_21.variants.lower Erdős and Lovász [ErLo75] proved that $\frac{8}{3}n-3\leq f(n)$ for all $n$ sorry line 90 source
Source documentation
Erdős and Lovász [ErLo75] proved that $\frac{8}{3}n-3\leq f(n)$ for all $n$.

depends on sorry what this means
Statement uses: Nat, LE.le, Real, HSub.hSub, HMul.hMul, HDiv.hDiv, OfNat.ofNat, Erdos21.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_21.variants.lower (n : ℕ) : (8 / 3 : ℝ) * n - 3 ≤ f n := by
…
```
Mentions: Erdos21.f
theorem Erdos21.erdos_21.variants.three_n It has been speculated (see e.g. [Ka94]) that the correct answer is $3n+O(1)$ sorry line 95 source
Source documentation
It has been speculated (see e.g. [Ka94]) that the correct answer is $3n+O(1)$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Filter.Eventually, And, LE.le, Erdos21.f, HAdd.hAdd, HMul.hMul, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_21.variants.three_n : answer(sorry) ↔ ∃ C : ℕ, ∀ᶠ n : ℕ in atTop,
    f n ≤ 3 * n + C ∧ 3 * n ≤ f n + C := by
…
```
Mentions: Erdos21.f
theorem Erdos21.erdos_21.variants.small_values $f(1)=1$, $f(2)=3$, $f(3)=6$ [Tr14], $f(4)=9$ [Tr14] and $f(5)=13$ [BaWa21] sorry line 101 source
Source documentation
$f(1)=1$, $f(2)=3$, $f(3)=6$ [Tr14], $f(4)=9$ [Tr14] and $f(5)=13$ [BaWa21].

depends on sorry what this means
Statement uses: And, Eq, Nat, Erdos21.f, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_21.variants.small_values : f 1 = 1 ∧ f 2 = 3 ∧ f 3 = 6 ∧ f 4 = 9 ∧ f 5 = 13 := by
…
```
Mentions: Erdos21.f