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erdős #139 · Szemerédi's theorem

← #138 · #140 → (packet.json; erdosproblems.com)

Let $r_{k} (N)$ be the size of the largest subset of ${1, \dots, N}$ which does not contain a non-trivial $k$ -term arithmetic progression. Prove that $r_{k} (N) = o (N)$ .

Worked, still open.

additive combinatorics · solved · prize $1000 · formalized (Lean) · 0 attempts

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

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formal

AMS 5 11 · solved (literature)

theorem erdos_139 (k : ℕ) (hk : 1 < k) :
    Filter.Tendsto (fun N => (r k N / N : ℝ)) Filter.atTop (𝓝 0)

formal-conjectures/139.lean ↗

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

links

Rothe-Ille · reference

A \subseteq N

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

A

contain arbitrarily long arithmetic progressions?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}

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}

k

-term arithmetic progression? Is it true that

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

status

solved

finding.noted · reviewer:will-blair · 1 day

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