Erdős Problems

1,217 source-owned questions · 604 with a formal statement · searchable by statement, number, topic and source status.

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Source status, exact formal material, and reviewed Results are separate signals.

Open per source608
Resolved per source556
Other source status53

Exact formal statement available604 / 1,217

With Repository-reviewed evidence2 / 1,217

31 Problems

Number	Question	Source says	Formal	Result here
#14	Let $A ⊆ \mathbb{N}$ . Let $B ⊆ \mathbb{N}$ 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 $\epsilon > 0$ and large $N$ , $\|\{1,\ldots,N\} \setminus B\| \gg_\epsilon N^{1/2 - \epsilon}$ ?openFormalized	open	In Lean	No Result decision
#30	Is it true that, for every $\varepsilon > 0$ , h(N) $= \sqrt N + O_{\varespilon}(N^\varespilon$ )openFormalized	open	In Lean	No Result decision
#39	Is there an infinite Sidon set $A\subset \mathbb{N}$ such that $\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$ for all $\varepsilon > 0$ ?openFormalized	open	In Lean	No Result decision
#41	Let `A ⊆ ℕ` be an infinite set such that the triple sums `a + b + c` are all distinct for `a, b, c` in `A` (aside from the trivial coincidences). Is it true that `liminf n → ∞ \|A ∩ {1, …, N}\| / N^(1/3) = 0`?openFormalized	open	In Lean	No Result decision
#42	Erdős Problem 42: Let M ≥ 1 and N be sufficiently large in terms of M. Is it true that for every maximal Sidon set `A ⊆ {1,…,N}` there is another Sidon set `B ⊆ {1,…,N}` of size M such that `(A - A) ∩ (B - B) = {0}`?solved (Lean)Formalized	solved (Lean)	In Lean	No Result decision
#43	If $A$ and $B$ are Sidon sets in $\{1,\ldots,N\}$ with $(A-A)\cap(B-B)=\{0\}$ , is it true that $\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+O(1)?$ disprovedFormalized	disproved	In Lean	No Result decision
#44	Erdős Problem 44: Let N ≥ 1 and `A ⊆ {1,…,N}` be a Sidon set. Is it true that, for any ε > 0, there exist M = M(ε) and `B ⊆ {N+1,…,M}` such that `A ∪ B ⊆ {1,…,M}` is a Sidon set of size at least `(1−ε)M^{1/2}`?openFormalized	open	In Lean	No Result decision
#152	Must `lim f n = ∞`?provedFormalized	proved	In Lean	No Result decision
#153	Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$ . Is it true that $\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty$ as $\lvert A\rvert\to \infty$ ?openFormalized	open	In Lean	No Result decision
#154	Let $A\subset \{1,\ldots,N\}$ be a Sidon set with $\lvert A\rvert\sim N^{1/2}$ . Must $A+A$ be well-distributed over all small moduli? In particular, must about half the elements of $A+A$ be even and half odd?proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#155	Is it true that for every $k \geq 1$ we have $F(N + k) \leq F(N) + 1$ for all sufficiently large $N$ ?openFormalized	open	In Lean	No Result decision
#156	Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$ ?openFormalized	open	In Lean	No Result decision
#157	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#158	A set is `B₂[1]` iff it is Sidon.openFormalized	open	In Lean	No Result decision
#198	The statement for which Baumgartner actually writes a proof.disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#241	Is it true that $f(N)\sim N^{1/3}$ ?openFormalized	open	In Lean	No Result decision
#340	Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a + b = c + d$ ). What is the order of growth of $A$ ? Is it true that $\|A \cap \{1, \ldots, N\}\| \gg N^{1/2 - \varepsilon}$ for all $\varepsilon > 0$ and large $N$ ?openFormalized	open	In Lean	No Result decision
#425	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#530	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#707	Erdős Problem 707: It is false that any finite Sidon set can be embedded in a perfect different set modulo some $n$ .disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#757	What is the supremum of the set of admissible numbers?openFormalized	open	In Lean	No Result decision
#772	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#773	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#840	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#861	No statement retained — open to read what the source holdssolvedNo formal declaration	solved	No formal statement	No Result decision
#862	Let $A_1(N)$ be the number of maximal Sidon subsets of $\{1,\ldots,N\}$ . Is it true that $A_1(N) < 2^{o(N^{1/2})}?$ solved (Lean)Formalized	solved (Lean)	In Lean	No Result decision
#863	No statement retained — open to read what the source holdsprovedNo formal declaration	proved	No formal statement	No Result decision
#864	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1191	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1194	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision
#1206	No statement retained — open to read what the source holdsopenNo formal declaration	open	No formal statement	No Result decision

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