erdős #3

← #2 · #4 → (packet.json; erdosproblems.com)

If $A \subseteq N$ has $\sum_{n \in A} \frac{1}{n} = \infty$ then must $A$ contain arbitrarily long arithmetic progressions?

unreviewedOpen. Worked here; no verified result yet.

number theory · open · prize $5000 · formalized (Lean) · 0 attempts

machinery: arithmetic-progressions,additive-combinatorics,Szemeredi-density-bound,prime-distribution

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Evidence

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

This is **exactly** the (still open) **Erdős conjecture on arithmetic progressions** (often called the **Erdős–Turán conjecture**):

candidate solution ↗

llm-hunter · codex 5.2 extra high, gpt 5.2, gpt pro 5.2 · unverified

4 LLM attack(s) recorded (codex 5.2 extra high, gpt 5.2, gpt pro 5.2); unverified.

candidate solution ↗

Formal proof

AMS 11 · open (literature)

theorem erdos_3 : answer(sorry) ↔ ∀ A : Set ℕ,
    (¬ Summable fun a : A ↦ 1 / (a : ℝ)) →
    ∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength k

formal-conjectures/3.lean ↗

OEIS4

A003002 — Size of the largest subset of the numbers [1...n] which does not contain a 3-term arithmetic progression.0,1,2,2,3,4,4,4,4,5,5,6,6,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11,11,11,12,12,13,13,13,13,14,14,14,14,15,16,16,16,16,16,16,16,A003003 — Size of the largest subset of the numbers [1...n] which doesn't contain a 4-term arithmetic progression.1,2,3,3,4,5,5,6,7,8,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,17,17,18,18,18,19,20,20,20,21,21,21,22,22,22,23,23,24 A003004 — Size of the largest subset of the numbers [1..n] which does not contain a 5-term arithmetic progression.1,2,3,4,4,5,6,7,8,8,9,10,11,12,12,13,14,15,16,16,16,16,16,17,18,18,19,20,21,21,22,22,23,24,24,25,26,27,28,28,29,30,31,32 A003005 — Size of the largest subset of the numbers [1..n] which doesn't contain a 6-term arithmetic progression.1,2,3,4,5,5,6,7,8,9,9,10,11,12,13,13,14,15,16,17,17,18,19,20,21,22,22,22,23,23,23,24,25,25,26,27,28,28,29,30,31,31,31,32

Connections

#139Let

r_{k} (N)

be the size of the largest subset of

{1, \dots, N}

which does not contain a non-trivial

k

-term arithmetic progression. Prove that

r_{k} (N) = o (N)

.A003002 #140Let

r_{3} (N)

be the size of the largest subset of

{1, \dots, N}

which does not contain a non-trivial

3

-term arithmetic progression. Prove that

r_{3} (N) ≪ N / (lo g N)^{C}

for every

C > 0

.A003002 #142Let

r_{k} (N)

be the largest possible size of a subset of

{1, \dots, N}

that does not contain any non-trivial

k

-term arithmetic progression. Prove an asymptotic formula for

r_{k} (N)

.A003002 #201Let

G_{k} (N)

be such that any set of

N

integers contains a subset of size at least

G_{k} (N)

which does not contain a

k

-term arithmetic progression. Determine the size of

G_{k} (N)

. How does it relate to

R_{k} (N)

, the size of the largest subset of

{1, \dots, N}

without a

k

-term arithmetic progression? Is it true that

N \to \infty lim \frac{R _{3} ( N )}{G _{3} ( N )} = 1 ?

A003002

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