Erdős Problem 361

Reference: erdosproblems.com/361

theorem Erdos361.erdos_361.bigO (c : ℝ) (hc : 0 < c) (A : ℕ → ℕ) (hA : ∀ (c n : ℕ), A n = {B ∈ (Finset.Icc 1 ⌊c * n⌋₊).powerset | n ≠ ∑ a ∈ B, a}.sup Finset.card) : (fun (n : ℕ) => ↑(A n)) ≪ sorry

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?

theorem Erdos361.erdos_361.bigTheta (c : ℝ) (hc : 0 < c) (A : ℕ → ℕ) (hA : ∀ (c n : ℕ), A n = {B ∈ (Finset.Icc 1 ⌊c * n⌋₊).powerset | n ≠ ∑ a ∈ B, a}.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 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?

theorem Erdos361.erdos_361.smallO (c : ℝ) (hc : 0 < c) (A : ℕ → ℕ) (hA : ∀ (c n : ℕ), A n = {B ∈ (Finset.Icc 1 ⌊c * n⌋₊).powerset | n ≠ ∑ a ∈ B, a}.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 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?