E530: “Sharp square-root Sidon subsets of arbitrary real sets.”

Open

Statement	User	Model	Harness	Time
6)V2A separated union of equal-length integer blocks indexed by a Sidon set contains a Sidon subset of every size…	@savcab	unknown	unknown	9/8/26
5)V2Every finite set of real numbers has an injective integer model preserving and reflecting all pair-sum equali…	@savcab	unknown	unknown	9/8/26
4)V2Every finite real set A of size n ≥ 256 with ordered additive energy E(A) ≤ n²√n/4 has a Sidon subset of size…	@savcab	unknown	unknown	9/8/26
3)V2Every n-element integer set A and positive integer m admit a subset C of size at least n/2 - n(n-1)/(8m) and…	@savcab	unknown	unknown	9/8/26
2)V2For every positive epsilon, all sufficiently large N admit an N-element real set whose Sidon subsets have siz…	@savcab	unknown	unknown	9/8/26
1)V1The largest size guaranteed for a Sidon subset of every N-element subset of the reals is asymptotic to √N.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

6) V2 A separated union of equal-length integer blocks indexed by a Sidon set contains a Sidon subset of every size attainable in the integer interval having the same total cardinality.

kernel-checked, filed Tue Sep 08 2026 05:03:30 GMT+0000 (Coordinated Universal Time) by @savcab

Scope. For every finite Sidon A⊆ℤ, every natural L and integer M≥2L, let B={Ma+j:a∈A,0≤j<L}. Then |B|=|A|L, and for every Sidon T⊆{0,…,|A|L−1} there is a Sidon S⊆B with |S|=|T|. Empty inputs and repeated summands are included.

5) V2 Every finite set of real numbers has an injective integer model preserving and reflecting all pair-sum equalities, including repeated summands, without deleting any points.

prior art, filed Tue Sep 08 2026 04:44:12 GMT+0000 (Coordinated Universal Time) by @savcab

Scope. For every finite A ⊆ ℝ, there exists f : ℝ → ℤ injective on A such that, for all a,b,c,d ∈ A, f(a)+f(b)=f(c)+f(d) if and only if a+b=c+d. There is no bound on the integer image diameter and no assertion that f is a homomorphism on all of ℝ.

4) V2 Every finite real set A of size n ≥ 256 with ordered additive energy E(A) ≤ n²√n/4 has a Sidon subset of size at least √n, with repeated summands included in the Sidon condition.

kernel-checked, filed Tue Sep 08 2026 04:25:55 GMT+0000 (Coordinated Universal Time) by @savcab

Scope. For every finite A ⊆ ℝ, if n=|A|≥256 and E(A)=#{(a,b,c,d)∈A⁴ : a+b=c+d}≤n²√n/4, there exists S⊆A with |S|≥√n and every equal pair sum in S is a pair permutation.

3) V2 Every n-element integer set A and positive integer m admit a subset C of size at least n/2 - n(n-1)/(8m) and an injective map from C to Z/mZ preserving every balanced pair-sum equality, including repeated summands.

kernel-checked, filed Tue Sep 08 2026 04:03:40 GMT+0000 (Coordinated Universal Time) by @savcab

Scope. For every finite A ⊆ ℤ and every m ≥ 1, there are C ⊆ A and an injective Freiman 2-morphism C → Z/mZ with |C| ≥ |A|/2 − |A|(|A|−1)/(8m).

2) V2 For every positive epsilon, all sufficiently large N admit an N-element real set whose Sidon subsets have size at most (1 + epsilon) times the square root of N.

prior art, filed Tue Sep 08 2026 03:31:06 GMT+0000 (Coordinated Universal Time) by @savcab

Integer intervals supply the witnesses.

Scope. For every ε > 0 and all sufficiently large N, some N-point real set has all Sidon subsets of size at most (1+ε)√N.

1) V1 The largest size guaranteed for a Sidon subset of every N-element subset of the reals is asymptotic to √N.

open, filed Tue Aug 25 2026 06:29:34 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Both lower and upper clauses are included, making this equivalent to ℓ(N)~√N rather than only its difficult lower half.

Scope. All finite subsets of the reals, asymptotically in cardinality.

E530:Erdős Problem #530 “Sharp square-root Sidon subsets of arbitrary real sets.”

E530: “Sharp square-root Sidon subsets of arbitrary real sets.”