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Erdős Problems

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

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  • Open per source608
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Problems

31 Problems

SelectNumberQuestionOpen
#14Let A⊆NA ⊆ \mathbb{N}. Let B⊆NB ⊆ \mathbb{N} be the set of integers which are representable in exactly one way as the sum of two elements from AA. Is it true that for all ϵ>0\epsilon > 0 and large NN, ∣{1,…,N}∖B∣≫ϵN1/2−ϵ|\{1,\ldots,N\} \setminus B| \gg_\epsilon N^{1/2 - \epsilon}?Source says openFormalized
#30Is it true that, for every ε>0\varepsilon > 0, h(N) Unparsed notation: = \sqrt N + O_{\varespilon}(N^\varespilon)Source says openFormalized
#39Is there an infinite Sidon set A⊂NA\subset \mathbb{N} such that ∣A∩{1…,N}∣≫ϵN1/2−ϵ\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon} for all ε>0\varepsilon > 0?Source says openFormalized
#41Let 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?Source says openFormalized
#42Erdős 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}?Source says solved (Lean)Formalized
#43If AA and BB are Sidon sets in {1,…,N}\{1,\ldots,N\} with (A−A)∩(B−B)={0}(A-A)\cap(B-B)=\{0\}, is it true that (∣A∣2)+(∣B∣2)≤(f(N)2)+O(1)?\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+O(1)?Source says disprovedFormalized
#44Erdős 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}?Source says openFormalized
#152Must lim f n = ∞?Source says provedFormalized
#153Let AA be a finite Sidon set and A+A={s1<⋯<st}A+A=\{s_1<\cdots<s_t\}. Is it true that 1t∑1≤i<t(si+1−si)2→∞\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty as ∣A∣→∞\lvert A\rvert\to \infty?Source says openFormalized
#154Let A⊂{1,…,N}A\subset \{1,\ldots,N\} be a Sidon set with ∣A∣∼N1/2\lvert A\rvert\sim N^{1/2}. Must A+AA+A be well-distributed over all small moduli? In particular, must about half the elements of A+AA+A be even and half odd?Source says proved (Lean)Formalized
#155Is it true that for every k≥1k \geq 1 we have F(N+k)≤F(N)+1 F(N + k) \leq F(N) + 1 for all sufficiently large NN?Source says openFormalized
#156Does there exist a maximal Sidon set A⊂{1,…,N}A\subset \{1,\ldots,N\} of size O(N1/3)O(N^{1/3})?Source says openFormalized
#157No statement retained — open to read what the source holdsSource says provedNo formal declaration
#158A set is B₂[1] iff it is Sidon.Source says openFormalized
#198The statement for which Baumgartner actually writes a proof.Source says disproved (Lean)Formalized
#241Is it true that f(N)∼N1/3f(N)\sim N^{1/3}?Source says openFormalized
#340Let A={1,2,4,8,13,21,31,45,66,81,97,…}A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\} be the greedy Sidon sequence: we begin with 11 and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to a+b=c+da + b = c + d). What is the order of growth of AA? Is it true that ∣A∩{1,…,N}∣≫N1/2−ε|A \cap \{1, \ldots, N\}| \gg N^{1/2 - \varepsilon} for all ε>0\varepsilon > 0 and large NN?Source says openFormalized
#425No statement retained — open to read what the source holdsSource says openNo formal declaration
#530No statement retained — open to read what the source holdsSource says openNo formal declaration
#707Erdős Problem 707: It is false that any finite Sidon set can be embedded in a perfect different set modulo some nn.Source says disproved (Lean)Formalized
#757What is the supremum of the set of admissible numbers?Source says openFormalized
#772No statement retained — open to read what the source holdsSource says provedNo formal declaration
#773No statement retained — open to read what the source holdsSource says openNo formal declaration
#840No statement retained — open to read what the source holdsSource says openNo formal declaration
#861No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#862Let A1(N)A_1(N) be the number of maximal Sidon subsets of {1,…,N}\{1,\ldots,N\}. Is it true that A1(N)<2o(N1/2)?A_1(N) < 2^{o(N^{1/2})}?Source says solved (Lean)Formalized
#863No statement retained — open to read what the source holdsSource says provedNo formal declaration
#864No statement retained — open to read what the source holdsSource says openNo formal declaration
#1191No statement retained — open to read what the source holdsSource says openNo formal declaration
#1194No statement retained — open to read what the source holdsSource says openNo formal declaration
#1206No statement retained — open to read what the source holdsSource says openNo formal declaration