Erdős Problem 361

Reference: erdosproblems.com/361

theorem erdos_361.bigO (c : ℝ) (hc : 0 < c) (A : ℕ → ℕ) (hA : ∀ (c n : ℕ), A n = (Finset.filter (fun (B : Finset ℕ) => n ≠ ∑ a ∈ B, a) (Finset.Icc 1 ⌊c * n⌋₊).powerset).sup Finset.card) : (fun (n : ℕ) => ↑(A n)) =O[Filter.atTop] sorry

Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?

theorem erdos_361.bigTheta (c : ℝ) (hc : 0 < c) (A : ℕ → ℕ) (hA : ∀ (c n : ℕ), A n = (Finset.filter (fun (B : Finset ℕ) => n ≠ ∑ a ∈ B, a) (Finset.Icc 1 ⌊c * n⌋₊).powerset).sup Finset.card) : (fun (n : ℕ) => ↑(A n)) =Θ[Filter.atTop] sorry

Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?

theorem erdos_361.smallO (c : ℝ) (hc : 0 < c) (A : ℕ → ℕ) (hA : ∀ (c n : ℕ), A n = (Finset.filter (fun (B : Finset ℕ) => n ≠ ∑ a ∈ B, a) (Finset.Icc 1 ⌊c * n⌋₊).powerset).sup Finset.card) : (fun (n : ℕ) => ↑(A n)) =o[Filter.atTop] sorry

Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?