erdős #201

← #200 · #202 → (packet.json; erdosproblems.com)

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

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additive combinatorics · open · 0 attempts

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

Write, for a finite set (A\subset\mathbb Z), [ r_k(A):=\max{|B|:B\subseteq A\text{ and }B\text{ contains no }k\text{-term AP}}. ] Then your quantity is [ G_k(N)=\min_{\substack{A\subset\mathbb Z\ |A|=N}} r_k(A), ] while [ R_k(N)=r_k({1,2,\dots,N}). ]

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llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

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oeis

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

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#3If

A \subseteq N

has

\sum_{n \in A} \frac{1}{n} = \infty

then must

A

contain arbitrarily long arithmetic progressions?A003002 #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

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open

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