erdős #530

← #529 · #531 → (packet.json; erdosproblems.com)

Let $ℓ (N)$ be maximal such that in any finite set $A \subset R$ of size $N$ there exists a Sidon subset $S$ of size $ℓ (N)$ (i.e. the only solutions to $a + b = c + d$ in $S$ are the trivial ones). Determine the order of $ℓ (N)$ .In particular, is it true that $ℓ (N) \sim N^{1/2}$ ?

Worked, still open.

number theory · open · 0 attempts

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

Write [ s(A):=\max{|S|:S\subseteq A\ \text{is Sidon}}, \qquad \ell(N)=\min_{|A|=N}s(A). ] This is the usual “worst–case” size of the largest Sidon subset. In the integer setting this function is commonly denoted $g(n)$.

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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

A143824 — Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences |x(i)-x(j)| are distinct.0,1,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10

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Vela Sidon frontier (A309370) · verified work

A B₂/Sidon problem — the same object family as Vela's verified Sidon records, where nine improved terms were accepted into OEIS A309370 (the campaign's first external adoption).

OEIS A309370 ↗ · verified-combinatorics (witnesses + verify.py) ↗ · the Erdős campaign

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#14Let

A \subseteq N

. Let

B \subseteq N

be the set of integers which are representable in exactly one way as the sum of two elements from

A

.Is it true that for all

ϵ > 0

and large

N

∣{1, \dots, N} \ B ∣ ≫_{ϵ} N^{1/2 - ϵ} ?

Is it possible that

∣{1, \dots, N} \ B ∣ = o (N^{1/2})?

A143824 #30Let

h (N)

be the maximum size of a Sidon set in

{1, \dots, N}

. Is it true that, for every

ϵ > 0

h (N) = N^{1/2} + O_{ϵ} (N^{ϵ})?

A143824 #43If

A, B \subset {1, \dots, N}

are two Sidon sets such that

(A - A) \cap (B - B) = {0}

then is it true that

(2 ∣ A ∣) + (2 ∣ B ∣) \leq (2 f ( N )) + O (1),

where

f (N)

is the maximum possible size of a Sidon set in

{1, \dots, N}

? If

∣ A ∣ = ∣ B ∣

then can this bound be improved to

(2 ∣ A ∣) + (2 ∣ B ∣) \leq (1 - c + o (1)) (2 f ( N ))

for some constant

c > 0

?A143824 #155Let

F (N)

be the size of the largest Sidon subset of

{1, \dots, N}

. Is it true that for every

k \geq 1

we have

F (N + k) \leq F (N) + 1

for all sufficiently large

N

?A143824 #861Let

f (N)

be the size of the largest Sidon subset of

{1, \dots, N}

and

A (N)

be the number of Sidon subsets of

{1, \dots, N}

. Is it true that

A (N) / 2^{f (N)} \to \infty ?

Is it true that

A (N) = 2^{(1 + o (1)) f (N)} ?

A143824

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