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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

110 Problems · 1 with reviewed evidence

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NumberQuestionOpen
#529No statement retained — open to read what the source holdsopenNo formal declaration
#588No statement retained — open to read what the source holdsopenNo formal declaration
#589No statement retained — open to read what the source holdsopenNo formal declaration
#604No statement retained — open to read what the source holdsopenNo formal declaration
#605No statement retained — open to read what the source holdsprovedNo formal declaration
#606No statement retained — open to read what the source holdssolvedNo formal declaration
#607No statement retained — open to read what the source holdsprovedNo formal declaration
#633Which triangles can only be decomposed into a square number of congruent triangles?solvedFormalized
#634No statement retained — open to read what the source holdsopenNo formal declaration
#651No statement retained — open to read what the source holdsdisprovedNo formal declaration
#652No statement retained — open to read what the source holdsprovedNo formal declaration
#653Let x1,…,xn∈R2x_1,\ldots,x_n\in \mathbb{R}^2 and let R(xi)=#{∣xj−xi∣:j≠i}R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\}, where the points are ordered such that R(x1)≤⋯≤R(xn).R(x_1)\leq \cdots \leq R(x_n). Let g(n)g(n) be the maximum number of distinct values the R(xi)R(x_i) can take. Is it true that g(n)≥(1−o(1))ng(n) \geq (1-o(1))n?openFormalized
#654No statement retained — open to read what the source holdsopenNo formal declaration
#655Let x1,…,xn∈R2x_1,\ldots,x_n\in \mathbb{R}^2 be such that no circle whose centre is one of the xix_i contains three other points. Are there at least (1+c)n2(1+c)\frac{n}{2} distinct distances determined between the xix_i, for some constant c>0c>0 and all nn sufficiently large?openFormalized
#657No statement retained — open to read what the source holdsopenNo formal declaration
#659Is there a set of nn points in R2\mathbb{R}^2 such that every subset of 44 points determines at least 33 distances, yet the total number of distinct distances is ≪nlog⁡n\ll \frac{n}{\sqrt{\log n}}?proved (Lean)Formalized
#660No statement retained — open to read what the source holdsopenNo formal declaration
#661No statement retained — open to read what the source holdsopenNo formal declaration
#662No statement retained — open to read what the source holdsopenNo formal declaration
#668No statement retained — open to read what the source holdsopenNo formal declaration
#669No statement retained — open to read what the source holdsopenNo formal declaration
#670No statement retained — open to read what the source holdsopenNo formal declaration
#704No statement retained — open to read what the source holdsopenNo formal declaration
#733No statement retained — open to read what the source holdsprovedNo formal declaration
#735No statement retained — open to read what the source holdssolvedNo formal declaration
#754No statement retained — open to read what the source holdsprovedNo formal declaration
#755Erdős asked whether every nn-point set in R6\mathbb{R}^6 spans at most (1/27+o(1))n3(1/27 + o(1)) n^3 unit equilateral triangles.provedFormalized
#756Let A⊂R2A\subset \mathbb{R}^2 be a set of nn points. Can there be ≫n\gg n many distinct distances each of which occurs for more than nn many pairs from AA?proved (Lean)Formalized
#757What is the supremum of the set of admissible numbers?openFormalized
#769Let c(n)c(n) be minimal such that if k≥c(n)k\geq c(n) then the nn-dimensional unit cube can be decomposed into kk homothetic nn-dimensional cubes. Give good bounds for c(n)c(n) — in particular, is it true that c(n)≫nnc(n)\gg n^n?openFormalized
#798Let t(n)t(n) be the minimum number of points in {1,…,n}2\{1,\ldots,n\}^2 such that the (t2)\binom{t}{2} lines determined by these points cover all points in {1,…,n}2\{1,\ldots,n\}^2.proved (Lean)Formalized
#827No statement retained — open to read what the source holdsopenNo formal declaration
#831No statement retained — open to read what the source holdsopenNo formal declaration
#838No statement retained — open to read what the source holdsopenNo formal declaration
#846Erdős Problem 846 Let A ⊂ ℝ² be an infinite set for which there exists some ϵ>0 such that in any subset of A of size n there are always at least ϵn with no three on a line. Is it true that A is the union of a finite number of sets where no three are on a line?disproved (Lean)Formalized
#898If A,B,C∈R2A,B,C\in \mathbb{R}^2 form a triangle and PP is a point in the interior then, if NN is where the perpendicular from PP to ABAB meets the triangle, and similarly for MM and LL, PA‾+PB‾+PC‾≥2(PM‾+PN‾+PL‾). \overline{PA}+\overline{PB}+\overline{PC}\geq 2(\overline{PM}+\overline{PN}+\overline{PL}). proved (Lean)Formalized
#953No statement retained — open to read what the source holdsopenNo formal declaration
#956No statement retained — open to read what the source holdsopenNo formal declaration
#957No statement retained — open to read what the source holdsprovedNo formal declaration
#958Let A⊂R2A\subset \mathbb{R}^2 be a finite set of size nn, and let {d1,…,dk}\{d_1,\ldots,d_k\} be the set of distances determined by AA. Let f(d)f(d) be the multiplicity of dd, that is, the number of unordered pairs from AA of distance dd apart.disproved (Lean)Formalized
#959Let A⊆R2A\subseteq \mathbb{R}^2 be a set of size nn and let {d1,…,dk}\{d_1,\ldots,d_k\} be the set of distinct distances determined by AA. Let f(d)f(d) be the number of times the distance dd is determined, ordered so that f(d1)≥f(d2)≥⋯≥f(dk)f(d_1)\geq f(d_2)\geq \cdots \geq f(d_k). Estimate max⁡(f(d1)−f(d2)),\max (f(d_1)-f(d_2)), where the maximum is taken over all AA of size nn (this is extremalGap n).openFormalized
#960No statement retained — open to read what the source holdsdisprovedNo formal declaration
#982If nn distinct points in R2\mathbb{R}^2 form a convex polygon then some vertex has at least ⌊n2⌋\lfloor\frac{n}{2}\rfloor different distances to other vertices.falsifiableFormalized
#1069No statement retained — open to read what the source holdssolvedNo formal declaration
#1070No statement retained — open to read what the source holdsopenNo formal declaration
#1071Can a finite set of disjoint unit segments in a unit square be maximal? Solved affirmatively by [Da85], who gave an explicit construction.proved (Lean)Formalized
#1082Let A⊂R2A\subset \mathbb{R}^2 be a set of nn points with no three on a line. Does AA determine at least ⌊n/2⌋\lfloor n/2\rfloor distinct distances?falsifiableFormalized
#1083No statement retained — open to read what the source holdsopenNo formal declaration

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