703.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 703

References:

erdosproblems.com/703
[Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.
[Er76b] Erdős, P., Problems and results in graph theory and combinatorial analysis. Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975) (1976), 169-192.
[Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
[Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.
[FrFu84b] Frankl, P. and Füredi, Z., On hypergraphs without two edges intersecting in a given number of vertices. J. Combin. Theory Ser. A (1984), 230-236.
[Fr77b] Frankl, P., An intersection problem for finite sets. Acta Math. Acad. Sci. Hungar. (1977), 371-373.
[FrRo87] Frankl, Peter and Rödl, Vojtech, Forbidden intersections. Trans. Amer. Math. Soc. (1987), 259-286.

Imports (1)

FormalConjecturesUtil

def Erdos703.T T n r is maximal such that there exists a family line 51 source
Source documentation
T n r is maximal such that there exists a family $\mathcal{F}$ of subsets of $\{1,\ldots,n\}$ of size T n r such that $\lvert A\cap B\rvert\neq r$ for all $A,B\in \mathcal{F}$ (including $A=B$).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, Fin, And, Membership.mem, Ne, Finset.card, Inter.inter, Eq
5 instances, coercions and auxiliary constants left out.
```
noncomputable def T (n r : ℕ) : ℕ :=
…
```
theorem Erdos703.erdos_703 sorry line 68 source
depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, And, Nat, HMul.hMul, HSub.hSub, HDiv.hDiv, Erdos703.T, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos703.erdos_703.variants.zero It is trivial that $T(n,0)=2^{n-1}$ sorry line 75 source
Source documentation
It is trivial that $T(n,0)=2^{n-1}$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, Erdos703.T, HPow.hPow, HSub.hSub
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_703.variants.zero (n : ℕ) (hn : 1 ≤ n) : T n 0 = 2 ^ (n - 1) := by
…
```
Mentions: Erdos703.T
theorem Erdos703.erdos_703.variants.frankl_furedi Frankl and Füredi [FrFu84b] proved that, for fixed sorry line 89 source
Source documentation
Frankl and Füredi [FrFu84b] proved that, for fixed $r$ and $n$ sufficiently large in terms of $r$, the maximal $T(n,r)$ is achieved by taking $$\mathcal{F} = \left\{ A\subseteq \{1,\ldots,n\} : \lvert A\rvert> \frac{n+r}{2}\textrm{ or } \lvert A\rvert < r\right\}$$ when $n+r$ is odd, and $$\mathcal{F} = \left\{ A\subseteq \{1,\ldots,n\} : \lvert A\backslash \{1\}\rvert\geq \frac{n+r}{2}\textrm{ or }\lvert A\rvert < r\right\}$$ when $n+r$ is even.

depends on sorry what this means
Statement uses: Nat, Filter.Eventually, Eq, Erdos703.T, ite, HMod.hMod, HAdd.hAdd, OfNat.ofNat, Finset.card, Finset, Fin, Finset.filter, Or, LT.lt, Real, HDiv.hDiv, Finset.univ, LE.le, Ne, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_703.variants.frankl_furedi (r : ℕ) : ∀ᶠ n : ℕ in Filter.atTop,
    T n r = if (n + r) % 2 = 1 then
      ((univ : Finset (Finset (Fin n))).filter fun A ↦
        (n + r : ℝ) / 2 < A.card ∨ A.card < r).card
    else
      ((univ : Finset (Finset (Fin n))).filter fun A ↦
        (n + r : ℝ) / 2 ≤ (A.filter fun i : Fin n ↦ (i : ℕ) ≠ 0).card ∨
          A.card < r).card := by
…
```
Mentions: Erdos703.T