# P2982: Erdős Problem 30 on Sidon Sets - ID: `P2982` - Reference: `erdos-problem-30` - Page: https://theoremdb.org/statements/P2982 - Record maturity: Reviewed problem with recorded work ## Problem 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 $\varepsilon > 0$. ### Context This problem concerns the classical extremal problem for Sidon sets, which are also known as $B_2$ sets in additive combinatorics. The quantity $h(N)$ represents the largest possible size of a Sidon set with elements bounded by $N$. It is known that $h(N) \sim \sqrt{N}$, and the problem asks for a refined understanding of the error term in this asymptotic formula. ### Problem setup - **Definition (A Sidon set).** A Sidon set is a set of natural numbers in which all sums of two (not necessarily distinct) elements are distinct; that is, if $a, b, c, d$ are elements of the set with $a \leq b$ and $c \leq d$, then $a + b = c + d$ implies $a = c$ and $b = d$. - **Definition (For functions $f, g : \mathbb{N} \to \mathbb{R}$, the notation $f(N) = O(g(N))$ as $N \to \infty$).** For functions $f, g : \mathbb{N} \to \mathbb{R}$, the notation $f(N) = O(g(N))$ as $N \to \infty$ means there exists a constant $C > 0$ and a threshold $N_0$ such that $|f(N)| \leq C \cdot |g(N)|$ for all $N \geq N_0$. - **Definition (The notation $f(N) = O_{\varepsilon}(g(N))$ indicates that the implied constant $C$ may depend on the parameter $\varepsilon$).** The notation $f(N) = O_{\varepsilon}(g(N))$ indicates that the implied constant $C$ may depend on the parameter $\varepsilon$. - **Remark.** This problem concerns the classical extremal problem for Sidon sets, which are also known as $B_2$ sets in additive combinatorics. The quantity $h(N)$ represents the largest possible size of a Sidon set with elements bounded by $N$. It is known that $h(N) \sim \sqrt{N}$, and the problem asks for a refined understanding of the error term in this asymptotic formula. ### What counts as a solution - 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 $\varepsilon > 0$. ## Status OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 30 as open. The unresolved remainder is the full displayed statement. 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 $\varepsilon > 0$. [1](#reference-1) ## Work ### Evidence for the current status **Claim 1 (Current status and unresolved remainder).** OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 30 as open. The unresolved remainder is the full displayed statement. 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 $\varepsilon > 0$. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 30 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: 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 $\varepsilon > 0$. ### Background and intake notes - Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 30 as open. The unresolved remainder is the full displayed statement. - The maintained database entry for Erdős Problem 30 was open at commit 8138974387d9030542daabe67faaa33eff9356f8. - The pinned Formal Conjectures declaration was matched by problem number and source locator. - The controlled TheoremDB source corpus was checked for an already published record with the same slug. ### Open directions - **Route 1** (reported): 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 $\varepsilon > 0$. [1](#reference-1) ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-30`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1. Erdős Problems database, Problem 30, maintained status record. Erdős Problems record 30, checked 2026-08-01. Problem 30; status field and linked bibliography https://www.erdosproblems.com/30 - Also cited at Problem 30; status snapshot 8138974387d9030542daabe67faaa33eff9356f8 - Also cited at See dataset.references[0] for the exact external source and locator. - Also cited at Editorial research route recorded 2026-07-31 - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31 - Source use: original_summary - Records the current open status and links the literature attached to this exact numbered problem. - Source used to formulate or check the problem record. - Source used to assess the problem's recorded status. - For Erdős Problem 30 on Sidon Sets: Records the current open status and links the literature attached to this exact numbered problem. - Source named by the research packet. 2. Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 30 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31 - Source use: original_summary - Pins the maintained database snapshot used for this release's dated status decision. - Source used to assess the problem's recorded status. - For Erdős Problem 30 on Sidon Sets: Pins the maintained database snapshot used for this release's dated status decision. 3. Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 30. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/30.lean:L39; theorem erdos_30; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/30.lean#L39 - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31 - Source use: original_summary - Supplies the pinned formal declaration whose human-readable editorial statement is published here. - Source used to assess the problem's recorded status. - For Erdős Problem 30 on Sidon Sets: Supplies the pinned formal declaration whose human-readable editorial statement is published here.