789.lean`

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

10 declarations, 8 contain sorry, 0 declared as axioms

Erdős Problem 789

In this problem, a function $h : \mathbb{N} \to\mathbb{N}$ is defined maximally by some counting property.

The problem asks to estimate $h(n)$. This has been interpreted here as asking for $\Theta(h(n))$. The principal version includes answer(sorry) for an unknown function.

Straus [Str66] proved that $h(n) \ll \sqrt{n}$. Erdős [Er62c] and Choi [Ch74b] proved that $(n\log(n))^{1/3} \ll h(n)$. Korsky [Ko26] improved this to $h(n) \gg \sqrt{n\log\log n/\log n}$. The variants record these bounds and the open question whether $h(n) = \Theta(\sqrt{n})$. The file proves Korsky's bound, following [Ko26], and derives from it the Erdős–Choi bound and $h(n) \neq O((n\log(n))^{1/3})$.

References:

erdosproblems.com/789
[Str66] Straus, E. G., On a problem in combinatorial number theory. J. Math. Sci. (1966), 77--80.
[Er62c] Erdős, Pál, Some remarks on number theory. {III}. Mat. Lapok (1962), 28--38.
[Ch74b] Choi, S. L. G., On an extremal problem in number theory. J. Number Theory (1974), 105--111.
[Ko26] Korsky, S., A near-square-root bound for an additive problem of Erdős and Straus (2026). Proof claim.

Imports (1)

FormalConjecturesUtil

def Erdos789.IsSubsetSumSeparatingCard Given a non-negative integer line 56 source
Source documentation
Given a non-negative integer $n$, we say $m$ is a separating cardinality of subset sums if, for any set $A$ of $n$ integers, there is some $B\subseteq A$ of size $\geq m$ such that subset sums of $B$ can only ever coincide when the subsets have the same cardinality.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset, Int, Eq, Finset.card, Exists, And, LE.le, Finset.Nonempty, Finset.sum
5 instances, coercions and auxiliary constants left out.
```
def IsSubsetSumSeparatingCard (n m : ℕ) : Prop :=
…
```
def Erdos789.subsetSumThreshold The subset sum threshold $h(n)$, for each positive $n$, is the maximal separating cardinality of subset sums for $n$ line 62 source
Source documentation
The subset sum threshold $h(n)$, for each positive $n$, is the maximal separating cardinality of subset sums for $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Erdos789.IsSubsetSumSeparatingCard
1 instances, coercions and auxiliary constants left out.
```
noncomputable def subsetSumThreshold (n : ℕ): ℕ :=
…
```
theorem Erdos789.erdos_789 Documented declaration: erdos_789 sorry line 73 source
Source documentation
Let $h(n)$ be maximal such that if $A\subseteq \mathbb{Z}$ with $\lvert A\rvert=n$ then there is $B\subseteq A$ with $\lvert B\rvert \geq h(n)$ such that if $a_1+\cdots+a_r=b_1+\cdots+b_s$ with $a_i,b_i\in B$ then $r=s$.
Estimate $h(n)$.

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos789.subsetSumThreshold, sorryAx, Bool.false
Proof uses: Lean.Name, 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 erdos_789 :
    (fun n ↦ (subsetSumThreshold n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: Erdos789.subsetSumThreshold
theorem Erdos789.erdos_789.variants.sq Documented declaration: sq sorry line 85 source
Source documentation
Let $h(n)$ be maximal such that if $A\subseteq \mathbb{Z}$ with $\lvert A\rvert=n$ then there is $B\subseteq A$ with $\lvert B\rvert \geq h(n)$ such that if $a_1+\cdots+a_r=b_1+\cdots+b_s$ with $a_i,b_i\in B$ then $r=s$.
Is $h(n) = \Theta(\sqrt{n})$?

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos789.subsetSumThreshold, Real.sqrt
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 erdos_789.variants.sq :
    (fun n ↦ (subsetSumThreshold n : ℝ)) =Θ[atTop] fun n ↦ √n := by
…
```
Mentions: Erdos789.subsetSumThreshold
theorem Erdos789.erdos_789.variants.isBigO_sq Straus [Str66] proved that $h(n) \ll \sqrt{n}$ sorry line 91 source
Source documentation
Straus [Str66] proved that $h(n) \ll \sqrt{n}$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos789.subsetSumThreshold, Real.sqrt
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 erdos_789.variants.isBigO_sq :
    (fun n ↦ (subsetSumThreshold n : ℝ)) =O[atTop] fun n ↦ √n := by
…
```
Mentions: Erdos789.subsetSumThreshold
theorem Erdos789.erdos_789.variants.sq_isBigO By the solved variant erdos_789.variants.isBigO_sq, in order to prove erdos_789.variants.sq it suffices to show $\sqrt{n}=O(h(n))$ sorry line 98 source
Source documentation
By the solved variant erdos_789.variants.isBigO_sq, in order to prove erdos_789.variants.sq it suffices to show $\sqrt{n}=O(h(n))$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Real.sqrt, Erdos789.subsetSumThreshold
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 erdos_789.variants.sq_isBigO :
    (fun n : ℕ ↦ √n) =O[atTop] fun n ↦ (subsetSumThreshold n : ℝ) := by
…
```
Mentions: Erdos789.subsetSumThreshold
theorem Erdos789.erdos_789.variants.sqrt_loglog_div_log_isBigO sorry line 106 source
depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Real.sqrt, HDiv.hDiv, HMul.hMul, Real.log, Erdos789.subsetSumThreshold
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos789.erdos_789.variants.cube_root_linearithmic_isBigO sorry line 116 source
depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HPow.hPow, HMul.hMul, Real.log, HDiv.hDiv, OfNat.ofNat, Erdos789.subsetSumThreshold
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos789.erdos_789.variants.isBigO_cube_root_linearithmic sorry line 125 source
depends on sorry what this means
Statement uses: Not, Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos789.subsetSumThreshold, HPow.hPow, HMul.hMul, Real.log, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos789.erdos_789.variants.cube_root_linearithmic sorry line 134 source
depends on sorry what this means
Statement uses: Not, Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos789.subsetSumThreshold, HPow.hPow, HMul.hMul, Real.log, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.