Erdős Problems

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

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With Repository-reviewed evidence2 / 1,217

1,217 Problems · 2 with reviewed evidence

3/26

Number	Question	Source says	Formal	Result here
#97	Does every convex polygon have a vertex with no other 4 vertices equidistant from it?Source says falsifiableFormalized	falsifiable	In Lean	No Result decision
#98	Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$ , with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\to \infty$ ?Source says openFormalized	open	In Lean	No Result decision
#99	For sufficiently large n, is it the case that any set of n points with minimum distance $1$ that minimizes diameter must contain an equilateral triangle of side length 1?Source says openFormalized	open	In Lean	No Result decision
#100	Is the diameter of $A$ at least $Cn$ for some constant $C > 0$ ?Source says openFormalized	open	In Lean	No Result decision
#101	Given $n$ points in $\mathbb{R}^2$ , no five of which are on a line, the number of lines containing four points is $o(n^2)$ .Source says openFormalized	open	In Lean	No Result decision
#102	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#103	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#104	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#105	Let $A,B\subset \mathbb{R}^2$ be disjoint sets of size $n$ and $n-3$ respectively, with not all of $A$ contained on a single line. Is there a line which contains at least two points from $A$ and no points from $B$ ?Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#106	No statement retained — open to read what the source holdsSource says falsifiableNo formal declaration	falsifiable	No formal statement	No Result decision
#107	Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$ , no three on a line, contain $n$ points which form the vertices of a convex $n$ -gon. Prove that $f(n) = 2^{n-2} + 1$ .Source says falsifiableFormalized	falsifiable	In Lean	No Result decision
#108	For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r) contains a subgraph of girth ≥ r and chromatic number ≥ k?Source says openFormalized	open	In Lean	No Result decision
#109	Any $A\subseteq \mathbb{N}$ of positive upper density contains a sumset $B+C$ where both $B$ and $C$ are infinite.Source says provedFormalized	proved	In Lean	No Result decision
#110	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#111	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#112	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#113	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#114	No statement retained — open to read what the source holdsSource says falsifiableNo formal declaration	falsifiable	No formal statement	No Result decision
#115	If $p(z)$ is a polynomial of degree $n$ such that $\{z : \lvert p(z)\rvert\leq 1\}$ is connected then is it true that $\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?$ Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#116	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#117	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#118	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#119	Is it true that $\limsup M_n = \infty$ ?Source says solvedFormalized	solved	In Lean	No Result decision
#120	Let $A \subseteq \mathbb{R}$ be an infinite set. Must there be a set $E \subseteq \mathbb{R}$ of positive measure which does not contain any set of the shape $a * A + b$ for some $a,b \in \mathbb{R}$ and $a \neq 0$ ?Source says openFormalized	open	In Lean	No Result decision
#121	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#122	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#123	Let $a, b, c$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^k b^l c^m$ ( $k, l, m ≥ 0$ ), none of which divide any other?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#124	Let $3\leq d_1 < d_2 < \cdots < d_r$ be integers such that all sufficiently large integers can be written as a sum of the shape $\sum_i c_ia_i$ where $c_i \in \{0, 1\}$ and $a_i$ has only the digits $0, 1$ when written in base $d_i$ . Then $\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.$ Source says openFormalized	open	In Lean	No Result decision
#125	Case 3: Does $A + B$ have positive upper and lower density that are equal? This is the literal interpretation of "positive density" which was falsified.Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#126	Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\|A\| = n$ then $\prod_{a\neq b\in A}(a + b)$ has at least $f(n)$ distinct prime factors. Is it true that $\frac{f(n)}{\log n} \to\infty$ ?Source says openFormalized	open	In Lean	No Result decision
#127	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#128	Let G be a graph with n vertices such that every induced subgraph on ≥ $n/2$ vertices has more than $n^2/50$ edges. Must G contain a triangle?Source says falsifiableFormalized	falsifiable	In Lean	No Result decision
#129	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#130	Let $A\subset\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in $A$ , where two vertices are joined by an edge if and only if they are an integer distance apart. How large can the chromatic number and clique number of this graph be? In particular, can the chromatic number be infinite?Source says openFormalized	open	In Lean	No Result decision
#131	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#132	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#133	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#134	Let $\epsilon,\delta>0$ and $n$ be sufficiently large in terms of $\epsilon$ and $\delta$ . Let $G$ be a triangle-free graph on $n$ vertices with maximum degree $<n^{1/2-\epsilon}$ . Can $G$ be made into a triangle-free graph with diameter $2$ by adding at most $\delta n^2$ edges?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#135	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#136	No statement retained — open to read what the source holdsSource says solvedNo formal declaration	solved	No formal statement	No Result decision
#137	Let $k\geq 3$ . Can the product of any $k$ consecutive integers $N$ ever be powerful? That is, must there always exist a prime $p\mid N$ such that $p^2\nmid N$ ?Source says openFormalized	open	In Lean	No Result decision
#138	In [Er80] Erdős asks whether $\lim_{k \to \infty} (W(k))^{1/k} = \infty$ Source says openFormalized	open	In Lean	No Result decision
#139	Erdős Problem 139: Let $r_k(N)$ be the size of the largest subset of ${1,...,N}$ which does not contain a non-trivial $k$ -term arithmetic progression. Prove that $r_k(N) = o(N)$ .Source says provedFormalized	proved	In Lean	No Result decision
#140	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#141	Let $k≥3$ . Are there $k$ consecutive primes in arithmetic progression?Source says openFormalized	open	In Lean	No Result decision
#142	Prove an asymptotic formula for $r_k(N)$ , the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$ -term arithmetic progression.Source says openFormalized	open	In Lean	No Result decision
#143	Does this imply that $\liminf \frac{\|A \cap [1,x]\|}{x} = 0?$ Source says openFormalized	open	In Lean	No Result decision
#144	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision