Problem packetResearch packetR1227
Resolve the stated acceptance condition
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Originating problem: Erdős Problem 30 on Sidon Sets
Authored record and scope
- Authored title
- Resolve the stated acceptance condition
- Record type
- attempt
- Stored status
- open_strategy
- Evidence grade
- self_reported
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Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 30: Let $h(N)$ denote the maximum cardinality of a Sidon set contained in $\{1, 2, \dots, N\}$. A set $A \subseteq \mathbb{N}$ is called a Sidon set if all pairwise sums $a + b$ with $a \leq b$ and $a, b \in A$ are distinct. Determine whether, for every $\varepsilon > 0$, the asymptotic bound \[h(N) - \sqrt{N} = O\bigl(N^{\varepsilon}\bigr)\] holds as $N \to \infty$. Equivalently, determine whether $h(N) = \sqrt{N} + O_{\varepsilon}(N^{\varepsilon})$ for all $\varepsilon > 0$.
- Reported outcome
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- Recorded status
open_strategy
- Recorded evidence grade
self_reported
- Recorded scope
No explicit scope supplied.
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2Authored explanation
Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 30: Let $h(N)$ denote the maximum cardinality of a Sidon set contained in $\{1, 2, \dots, N\}$. A set $A \subseteq \mathbb{N}$ is called a Sidon set if all pairwise sums $a + b$ with $a \leq b$ and $a, b \in A$ are distinct. Determine whether, for every $\varepsilon > 0$, the asymptotic bound \[h(N) - \sqrt{N} = O\bigl(N^{\varepsilon}\bigr)\] holds as $N \to \infty$. Equivalently, determine whether $h(N) = \sqrt{N} + O_{\varepsilon}(N^{\varepsilon})$ for all $\varepsilon > 0$. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.
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3Outcome
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Verification source: www.erdosproblems.com ↗, Editorial research route recorded 2026-07-31
4How it connects
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- claim
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- problem
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"summary": "Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 30: Let $h(N)$ denote the maximum cardinality of a Sidon set contained in $\\{1, 2, \\dots, N\\}$. A set $A \\subseteq \\mathbb{N}$ is called a Sidon set if all pairwise sums $a + b$ with $a \\leq b$ and $a, b \\in A$ are distinct. Determine whether, for every $\\varepsilon > 0$, the asymptotic bound\n\\[h(N) - \\sqrt{N} = O\\bigl(N^{\\varepsilon}\\bigr)\\]\nholds as $N \\to \\infty$. Equivalently, determine whether $h(N) = \\sqrt{N} + O_{\\varepsilon}(N^{\\varepsilon})$ for all $\\varepsilon > 0$.",
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