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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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1,217 Problems · 2 with reviewed evidence

11/26

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#481Let a1,…,ar,b1,…,br∈Na_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N} such that ∑i1ai>1\sum_{i}\frac{1}{a_i}>1. For any finite sequence of nn (not necessarily distinct) integers A=(x1,…,xn)A=(x_1,\ldots,x_n) let T(A)T(A) denote the sequence of length rnrn given by (aixj+bi)1≤j≤n,1≤i≤r.(a_ix_j+b_i)_{1\leq j\leq n, 1\leq i\leq r}. Prove that, if A1=(1)A_1=(1) and Ai+1=T(Ai)A_{i+1}=T(A_i), then there must be some AkA_k with repeated elements.proved (Lean)Formalized
#482No statement retained — open to read what the source holdssolvedNo formal declaration
#483No statement retained — open to read what the source holdsopenNo formal declaration
#484Prove that there exists an absolute constant c>0c>0 such that, whenever {1,…,N}\{1,\ldots,N\} is kk-coloured (and NN is large enough depending on kk) then there are at least cNcN many integers in {1,…,N}\{1,\ldots,N\} which are representable as a monochromatic sum (that is, a+ba+b where a,b∈{1,…,N}a,b\in \{1,\ldots,N\} are in the same colour class and a≠ba\neq b).proved (Lean)Formalized
#485No statement retained — open to read what the source holdsprovedNo formal declaration
#486For each n∈Nn \in \mathbb{N} choose some Xn⊆Z/nZX_n \subseteq \mathbb{Z}/n\mathbb{Z}. Let B={m∈N:∀n,m≢x(modn) for all x∈Xn}B = \{m \in \mathbb{N} : \forall n, m \not\equiv x \pmod{n} \text{ for all } x \in X_n\}. Must BB have a logarithmic density?openFormalized
#487Let A⊆NA\subseteq \mathbb{N} have positive density. Must there exist distinct a,b,c∈Aa,b,c\in A such that [a,b]=c[a,b]=c (where [a,b][a,b] is the least common multiple of aa and bb)?proved (Lean)Formalized
#488Let AA be a finite set and B={n≥1:a∣n for some a∈A}.B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}. Is it true that, for every m>n≥max⁡(A)m>n\geq \max(A), ∣B∩[1,m]∣m<2∣B∩[1,n]∣n?\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?falsifiableFormalized
#489Let A⊆NA\subseteq \mathbb{N} be a set such that ∣A∩[1,x]∣=o(x1/2)\lvert A\cap [1,x]\rvert=o(x^{1/2}). Let B={n≥1:a∤n for all a∈A}B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}. If B={b1<b2<⋯ }B=\{b_1 < b_2 < \cdots\} then is it true that lim⁡x→∞1x∑bi<x(bi+1−bi)2\lim_{x \to \infty} \frac{1}{x}\sum_{b_i < x}(b_{i+1}-b_i)^2 exists (and is finite)?openFormalized
#490No statement retained — open to read what the source holdsproved (Lean)No formal declaration
#491No statement retained — open to read what the source holdsprovedNo formal declaration
#492No statement retained — open to read what the source holdsdisprovedNo formal declaration
#493Does there exist a kk such that every sufficiently large integer can be written in the form ∏i=1kai−∑i=1kai\prod_{i=1}^k a_i - \sum_{i=1}^k a_i for some integers ai≥2a_i\geq 2?proved (Lean)Formalized
#494Selfridge and Straus [SeSt58] proved that AA is determined by AkA_k if ∣A∣|A| is divisible by a prime greater than kk.provedFormalized
#495Let α,β∈R\alpha,\beta \in \mathbb{R}. Is it true thatlim inf⁡n→∞n∥nα∥∥nβ∥=0\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0? This is also known as the Littlewood conjecture.openFormalized
#496No statement retained — open to read what the source holdsprovedNo formal declaration
#497How many antichains in [n][n] are there? That is, how many families of subsets of [n][n] are there such that, if F\mathcal{F} is such a family and A,B∈FA,B\in \mathcal{F}, then A⊈BA\not\subseteq B?solved (Lean)Formalized
#498Let z1,…,zn∈Cz_1,\ldots,z_n\in\mathbb{C} with 1≤∣zi∣1\leq \lvert z_i\rvert for 1≤i≤n1\leq i\leq n. Let DD be an arbitrary disc of radius 11. Is it true that the number of sums of the shape ∑i=1nϵizi for ϵi∈{−1,1}\sum_{i=1}^n\epsilon_iz_i \textrm{ for }\epsilon_i\in \{-1,1\} which lie in DD is at most (n⌊n/2⌋)\binom{n}{\lfloor n/2\rfloor}?proved (Lean)Formalized
#499Let MM be a real n×nn \times n doubly stochastic matrix. Does there exist some σ∈Snσ \in S_n such that ∏1≤i≤nMi,σ(i)≥n−n? \prod_{1 \leq i \leq n} M_{i, σ(i)} \geq n^{-n}? This is true, and was proved by Marcus and Minc [MaMi62]proved (Lean)Formalized
#500No statement retained — open to read what the source holdsopenNo formal declaration
#501For every x∈Rx \in \mathbb{R} let Ax⊂RA_x \subset \mathbb{R} be a bounded set with outer measure <1< 1. Must there exist an infinite independent set, that is, some infinite X⊆RX \subseteq \mathbb{R} such that x∉Ayx \notin A_y for all x≠y∈Xx \neq y \in X?openFormalized
#502What is the size of the largest A⊆RnA\subseteq \mathbb{R}^n such that there are only two distinct distances between elements of AA? That is, #{∣x−y∣:x≠y∈A}=2.\# \{ \lvert x-y\rvert : x\neq y\in A\} = 2.solved (Lean)Formalized
#503What is the size of the largest A⊆RnA \subseteq \mathbb{R}^n such that every three points from AA determine an isosceles triangle? That is, for any three points xx, yy, zz from AA, at least two of the distances ∣x−y∣|x - y|, ∣y−z∣|y - z|, ∣x−z∣|x - z| are equal.openFormalized
#504No statement retained — open to read what the source holdssolvedNo formal declaration
#505Erdős Problem 505 (disproved). Borsuk's conjecture is false for sufficiently large nn: there exists a dimension nn and a bounded set S⊆RnS \subseteq \mathbb{R}^n with positive diameter such that SS cannot be covered by n+1n + 1 subsets each of diameter strictly less than diam⁡(S)\operatorname{diam}(S).disproved (Lean)Formalized
#506No statement retained — open to read what the source holdsdecidableNo formal declaration
#507Let α(n)\alpha(n) be such that every set of nn points in the unit disk contains three points which determine a triangle of area at most α(n)\alpha(n). Estimate α(n)\alpha(n).openFormalized
#508The "chromatic number of the plane" is at least 4. This can be proven by considering the [Moser-Spindel graph](https://de.wikipedia.org/wiki/Moser-Spindel) or the [Golomb graph](https://en.wikipedia.org/wiki/Golomb_graph) graph.openFormalized
#509Let f(z)∈C[z]f(z) ∈ ℂ[z] be a monic non-constant polynomial. Can the set {z∈C:∣f(z)∣≤1}\{z ∈ ℂ : |f(z)| ≤ 1\} be covered by a set of closed discs the sum of whose radii is ≤2≤ 2?openFormalized
#510Chowla's cosine problemopenFormalized
#511No statement retained — open to read what the source holdsdisprovedNo formal declaration
#512Is it true that, if A⊂ZA\subset \mathbb{Z} is a finite set of size NN, then ∫01∣∑n∈Ae(nθ)∣dθ≫log⁡N,\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N, where e(x)=e2πixe(x)=e^{2\pi ix }?proved (Lean)Formalized
#513Let f be a transcendental entire function. What is the greatest possible value of liminf (fun r : ℝ => ratio r f) atTop?openFormalized
#514No statement retained — open to read what the source holdsopenNo formal declaration
#515No statement retained — open to read what the source holdsprovedNo formal declaration
#516Let f = ∑ aₖzⁿₖ be an entire function of finite order such that nₖ / k → ∞. Then limsup (fun r => ratio r f) atTop = 1. This is proved in [Fu63].provedFormalized
#517If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞, is it true that f assumes every value infinitely often?openFormalized
#518No statement retained — open to read what the source holdsprovedNo formal declaration
#519Let z1,…,zn∈Cz_1,\ldots,z_n\in \mathbb{C} with z1=1z_1=1. Must there exist an absolute constant c>0c>0 such that max⁡1≤k≤n∣∑izik∣>c? \max_{1\leq k\leq n}\left\lvert \sum_{i}z_i^k\right\rvert>c? proved (Lean)Formalized
#520Let ff be a Rademacher multiplicative function. Does there exist some constant c>0c > 0 such that, almost surely, lim sup⁡N→∞∑m≤Nf(m)Nlog⁡log⁡N=c? \limsup_{N \to \infty} \frac{\sum_{m \leq N} f(m)}{\sqrt{N \log \log N}} = c? openFormalized
#521Let (ϵk)k≥0(\epsilon_k)_{k\geq 0} be independently uniformly chosen at random from {−1,1}\{-1,1\}. If RnR_n counts the number of real roots of fn(z)=∑0≤k≤nϵkzkf_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k then is it true that, almost surely, lim⁡n→∞Rnlog⁡n=2π?\lim_{n\to \infty}\frac{R_n}{\log n}=\frac{2}{\pi}?openFormalized
#522Let f(z)=∑0≤k≤nϵkzkf(z)=\sum_{0\leq k\leq n} \epsilon_k z^k be a random polynomial, where ϵk∈{−1,1}\epsilon_k\in \{-1,1\} independently uniformly at random for 0≤k≤n0\leq k\leq n.openFormalized
#523No statement retained — open to read what the source holdsprovedNo formal declaration
#524No statement retained — open to read what the source holdsopenNo formal declaration
#525No statement retained — open to read what the source holdsprovedNo formal declaration
#526No statement retained — open to read what the source holdssolvedNo formal declaration
#527No statement retained — open to read what the source holdsprovedNo formal declaration
#528No statement retained — open to read what the source holdsopenNo formal declaration

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