erdős #14

← #13 · #15 → (packet.json; erdosproblems.com)

Let $A \subseteq N$ . 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 $∣{1, \dots, N} \ B ∣ = o (N^{1/2})?$

Open — best to date is a honest null, not yet sealed.

number theory · open · formalized (Lean) · 1 attempt

machinery: Sidon/B_h,additive-combinatorics,B_h-representation-function,consecutive-integer-window

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier 'sidon/B2' (transfer_strength=weak) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'weak'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

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

Let (r_A(n)) be the (unordered) representation function $$ r_A(n):= |\\{\\{a,a'\\}\subseteq A:\ a+a'=n\\}|, $$ so (B={n\in\mathbb N:\ r_A(n)=1}) and the “exceptional set” is [ E(N):=\bigl|\\{1,\dots,N\\}\setminus B\bigr|=|\\{n\le N:\ r_A(n)\ne 1\\}|. ] (If you instead count *ordered* representations, the questions are …

candidate solution ↗

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

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

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_14.parts.i :
    answer(sorry) ↔ ∀ A, ∀ ε > 0, nonUniqueSumCount A ≫ almostSquareRoot ε

formal-conjectures/14.lean ↗

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

links

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

#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 #530Let

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

S

are the trivial ones). Determine the order of

ℓ (N)

.In particular, is it true that

ℓ (N) \sim N^{1/2}

?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

status