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import FormalConjecturesUtilErdős Problem 361
open Filter
namespace Erdos361
Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq {1, \ldots, \lfloor c n \rfloor}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?
@[category research open, AMS 11]
theorem erdos_361.bigO
(c : ℝ) (hc : 0 < c)
(A : ℕ → ℕ)
(hA : ∀ c n, A n = ((Finset.Icc 1 ⌊c * n⌋₊).powerset.filter
(fun B ↦ n ≠ ∑ a ∈ B, a)).sup Finset.card) :
(fun n ↦ (A n : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ) := c:ℝhc:0 < cA:ℕ → ℕhA:∀ (c n : ℕ), A n = {B ∈ (Finset.Icc 1 ⌊c * n⌋₊).powerset | n ≠ ∑ a ∈ B, a}.sup Finset.card⊢ (fun n => ↑(A n)) =O[atTop] sorry
All goals completed! 🐙
Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq {1, \ldots, \lfloor c n \rfloor}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?
@[category research open, AMS 11]
theorem erdos_361.bigTheta
(c : ℝ) (hc : 0 < c)
(A : ℕ → ℕ)
(hA : ∀ c n, A n = ((Finset.Icc 1 ⌊c * n⌋₊).powerset.filter
(fun B ↦ n ≠ ∑ a ∈ B, a)).sup Finset.card) :
(fun n ↦ (A n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := c:ℝhc:0 < cA:ℕ → ℕhA:∀ (c n : ℕ), A n = {B ∈ (Finset.Icc 1 ⌊c * n⌋₊).powerset | n ≠ ∑ a ∈ B, a}.sup Finset.card⊢ (fun n => ↑(A n)) =Θ[atTop] sorry
All goals completed! 🐙
Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq {1, \ldots, \lfloor c n \rfloor}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?
@[category research open, AMS 11]
theorem erdos_361.smallO
(c : ℝ) (hc : 0 < c)
(A : ℕ → ℕ)
(hA : ∀ c n, A n = ((Finset.Icc 1 ⌊c * n⌋₊).powerset.filter
(fun B ↦ n ≠ ∑ a ∈ B, a)).sup Finset.card) :
(fun n ↦ (A n : ℝ)) =o[atTop] (answer(sorry) : ℕ → ℝ) := c:ℝhc:0 < cA:ℕ → ℕhA:∀ (c n : ℕ), A n = {B ∈ (Finset.Icc 1 ⌊c * n⌋₊).powerset | n ≠ ∑ a ∈ B, a}.sup Finset.card⊢ (fun n => ↑(A n)) =o[atTop] sorry
All goals completed! 🐙
end Erdos361