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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
  • Resolved per source556
  • Other source status53
Exact formal statement available604 / 1,217
With Repository-reviewed evidence2 / 1,217
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Problems

47 Problems

NumberQuestionOpen
#20Is it true that f(n,k)<cknf(n,k) < c_k^n for some constant ck>0c_k>0 and for all n>0n > 0?openFormalized
#21No statement retained — open to read what the source holdsprovedNo formal declaration
#83No statement retained — open to read what the source holdsprovedNo formal declaration
#120Let A⊆RA \subseteq \mathbb{R} be an infinite set. Must there be a set E⊆RE \subseteq \mathbb{R} of positive measure which does not contain any set of the shape a∗A+ba * A + b for some a,b∈Ra,b \in \mathbb{R} and a≠0a \neq 0?openFormalized
#161No statement retained — open to read what the source holdsopenNo formal declaration
#162No statement retained — open to read what the source holdsopenNo formal declaration
#171No statement retained — open to read what the source holdsprovedNo formal declaration
#191No statement retained — open to read what the source holdsprovedNo formal declaration
#192No statement retained — open to read what the source holdssolved (Lean)No formal declaration
#207No statement retained — open to read what the source holdsprovedNo formal declaration
#231No statement retained — open to read what the source holdsdisproved (Lean)No formal declaration
#447How large can a union-free collection F\mathcal{F} of subsets of [n][n] be? By union-free we mean there are no solutions to A∪B=CA\cup B=C with distinct A,B,C∈FA,B,C\in \mathcal{F}. Must ∣F∣=o(2n)\lvert \mathcal{F}\rvert =o(2^n)?proved (Lean)Formalized
#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
#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
#602Does every almost-disjoint family of countably infinite sets whose pairwise intersections all have size ≠ 1 have Property B?openFormalized
#603No statement retained — open to read what the source holdssolvedNo formal declaration
#624Let XX be a finite set of size nn and H(n)H(n) be such that there is a function f:{A:A⊆X}→Xf:\{A : A\subseteq X\}\to X so that for every Y⊆XY\subseteq X with ∣Y∣≥H(n)\lvert Y\rvert \geq H(n) we have {f(A):A⊆Y}=X\left\{ f(A) : A\subseteq Y\right\}=X. Prove that H(n)−log⁡2n→∞H(n)-\log_2 n \to \infty.openFormalized
#644No statement retained — open to read what the source holdsopenNo formal declaration
#664No statement retained — open to read what the source holdsdisprovedNo formal declaration
#665No statement retained — open to read what the source holdsopenNo formal declaration
#701Let F\mathcal{F} be a family of sets closed under taking subsets (i.e. if B⊆A∈FB\subseteq A\in\mathcal{F} then B∈FB\in \mathcal{F}). There exists some element xx such that whenever F′⊆F\mathcal{F}'\subseteq \mathcal{F} is an intersecting subfamily we have ∣F′∣≤∣{A∈F:x∈A}∣.\lvert \mathcal{F}'\rvert \leq \lvert \{ A\in \mathcal{F} : x\in A\}\rvert.openFormalized
#702No statement retained — open to read what the source holdsprovedNo formal declaration
#703No statement retained — open to read what the source holdsprovedNo formal declaration
#722No statement retained — open to read what the source holdsprovedNo formal declaration
#723If there is a finite projective plane of order nn then must nn be a prime power?falsifiableFormalized
#724No statement retained — open to read what the source holdsopenNo formal declaration
#725No statement retained — open to read what the source holdsopenNo formal declaration
#732No statement retained — open to read what the source holdsprovedNo formal declaration
#733No statement retained — open to read what the source holdsprovedNo formal declaration
#734No statement retained — open to read what the source holdsopenNo formal declaration
#747No statement retained — open to read what the source holdssolvedNo formal declaration
#776No statement retained — open to read what the source holdsopenNo formal declaration
#777No statement retained — open to read what the source holdssolvedNo formal declaration
#780No statement retained — open to read what the source holdsprovedNo formal declaration
#857Estimate m(n,k), or better give an asymptotic formula.openFormalized
#901No statement retained — open to read what the source holdsopenNo formal declaration
#903No statement retained — open to read what the source holdsprovedNo formal declaration
#1022Is there a constant ctc_t, where ct→∞c_t\to \infty as t→∞t\to \infty, such that if F\mathcal{F} is a finite family of finite sets, all of size at least tt, and for every set XX there are <ct∣X∣<c_t\lvert X\rvert many A∈FA\in \mathcal{F} with A⊆XA\subseteq X, then F\mathcal{F} has chromatic number 22 (in other words, has property B)?proved (Lean)Formalized
#1023Let F(n)F(n) be the maximal size of a family of subsets of {1,…,n}\{1,\ldots,n\} such that no set in this family is the union of other members of the family. Is it true that there is a constant c>0c>0 such that F(n)∼c2nn1/2?F(n)\sim c \frac{2^n}{n^{1/2}}?solved (Lean)Formalized
#1025No statement retained — open to read what the source holdssolvedNo formal declaration
#1026Let x1,…,xnx_1,\ldots,x_n be a sequence of distinct real numbers. Determine max⁡(∑xir), \max\left(\sum x_{i_r}\right), where the maximum is taken over all monotonic subsequences.solved (Lean)Formalized
#1027No statement retained — open to read what the source holdsprovedNo formal declaration
#1159No statement retained — open to read what the source holdsopenNo formal declaration
#1173No statement retained — open to read what the source holdsopenNo formal declaration
#1183No statement retained — open to read what the source holdsopenNo formal declaration

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