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import FormalConjecturesUtilErdős Problem 817
open Filter
namespace Erdos817Define $g_k(n)$ to be the minimal $N$ such that ${1, ..., N}$ contains some $A$ of size $|A| = n$ such that $$ \langle A\rangle = \left{\sum_{a \in A} \epsilon_a a : \epsilon_a \in{0, 1}\right} $$ contains no non-trivial $k$-term arithmetic progression.
noncomputable
def g (k : ℕ) (n : ℕ) : ℕ := sInf { N | ∃ A ⊆ Finset.Icc 1 N, A.card = n ∧
∀ s, s ⊆ { ∑ a ∈ B, a | B ∈ A.powerset } → s.IsAPOfLengthFree k}Let $k \geq 3$. Define $g_k(n)$ to be the minimal $N$ such that ${1, ..., N}$ contains some $A$ of size $|A| = n$ such that $$ \langle A\rangle = \left{\sum_{a \in A} \epsilon_a a : \epsilon_a \in{0, 1}\right} $$ contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that $$ g_3(n) \gg 3^n $$
-- Formalisation note : only formalising the "In particular" part
@[category research open, AMS 5 11]
theorem erdos_817 :
answer(sorry) ↔ (fun n => (3 ^ n : ℝ)) =O[atTop] fun n => (g 3 n : ℝ) := ⊢ True ↔ (fun n => 3 ^ n) =O[atTop] fun n => ↑(g 3 n)
All goals completed! 🐙A problem of Erdős and Sárközy who proved $$ g_3(n) \gg \frac{3^n}{n^{O(1)}}. $$
@[category research solved, AMS 5 11]
theorem erdos_817.variants.bdd_power : ∃ O > (0 : ℝ),
(fun (n : ℕ) => (3 ^ n : ℝ) / n ^ O) =O[atTop] fun n => (g 3 n : ℝ) := ⊢ ∃ O > 0, (fun n => 3 ^ n / ↑n ^ O) =O[atTop] fun n => ↑(g 3 n)
All goals completed! 🐙
end Erdos817