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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

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SelectNumberQuestionOpen
#1009No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1010No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1011No statement retained — open to read what the source holdsSource says openNo formal declaration
#1012No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1013No statement retained — open to read what the source holdsSource says openNo formal declaration
#1014Let R(k,l)R(k,l) be the Ramsey number, so the minimal nn such that every graph on at least nn vertices contains either a KkK_k or an independent set on ll vertices.Source says proved (Lean)Formalized
#1015No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1016No statement retained — open to read what the source holdsSource says openNo formal declaration
#1017No statement retained — open to read what the source holdsSource says openNo formal declaration
#1018No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1019No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1020No statement retained — open to read what the source holdsSource says falsifiableNo formal declaration
#1021No statement retained — open to read what the source holdsSource says provedNo 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)?Source says 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}}?Source says solved (Lean)Formalized
#1024No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1025No statement retained — open to read what the source holdsSource says solvedNo 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.Source says solved (Lean)Formalized
#1027No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1028Let H(n)=min⁡fmax⁡X⊆{1,…,n}∣∑x<y∈Xf(x,y)∣,H(n)=\min_f \max_{X\subseteq \{1,\ldots,n\}} \left\lvert \sum_{x<y\in X} f(x,y)\right\rvert, where ff ranges over all functions f:{1,…,n}2→{−1,1}f:\{1,\ldots,n\}^2\to \{-1,1\}. Estimate H(n)H(n).Source says solved (Lean)Formalized
#1029No statement retained — open to read what the source holdsSource says openNo formal declaration
#1030No statement retained — open to read what the source holdsSource says openNo formal declaration
#1031No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1032No statement retained — open to read what the source holdsSource says openNo formal declaration
#1033No statement retained — open to read what the source holdsSource says openNo formal declaration
#1034Let GG be a graph on nn vertices with >n2/4>n^2/4 many edges. Must there be a triangle TT in GG and vertices y1,…,yty_1,\ldots,y_t, where t>(12−o(1))nt>(\frac{1}{2}-o(1))n, such that every yiy_i is joined to at least two vertices of TT?Source says disproved (Lean)Formalized
#1035No statement retained — open to read what the source holdsSource says openNo formal declaration
#1036Let GG be a graph on nn vertices which does not contain a trivial (empty or complete) graph on more than clog⁡nc\log n vertices. Must GG contain at least 2Ωc(n)2^{\Omega_c(n)} many induced subgraphs which are not pairwise isomorphic?Source says proved (Lean)Formalized
#1037Let GG be a graph on nn vertices in which every degree occurs at most twice, and the number of distinct degrees is >(12+ϵ)n>(\frac{1}{2}+\epsilon)n. Must GG contain a trivial (empty or complete) subgraph of size 'much larger' than log⁡n\log n?Source says disproved (Lean)Formalized
#1038What is the infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1]?Source says openFormalized
#1039No statement retained — open to read what the source holdsSource says openNo formal declaration
#1040No statement retained — open to read what the source holdsSource says openNo formal declaration
#1041Let f(z)=∏i=1n(z−zi)∈C[x] f(z) = \prod_{i=1}^{n} (z - z_i) \in \mathbb{C}[x] with ∣zi∣<1|z_i| < 1 for all ii.Source says falsifiableFormalized
#1042No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1043Erdős Problem 1043: Let f∈C[x]f\in \mathbb{C}[x] be a monic polynomial. Must there exist a straight line ℓ\ell such that the projection of {z:∣f(z)∣≤1}\{ z: \lvert f(z)\rvert\leq 1\} onto ℓ\ell has measure at most 22?Source says disproved (Lean)Formalized
#1044Let f(z)=∏i=1n(z−zi)∈C[x]f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x] where ∣zi∣≤1\lvert z_i\rvert\leq 1 for all ii. If Λ(f)\Lambda(f) is the maximum of the lengths of the boundaries of the connected components of {z:∣f(z)∣<1} \{ z: \lvert f(z)\rvert<1\} then determine the infimum of Λ(f)\Lambda(f).Source says solved (Lean)Formalized
#1045No statement retained — open to read what the source holdsSource says openNo formal declaration
#1046No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#1047Let f∈C[x]f\in \mathbb{C}[x] be a monic polynomial with mm distinct roots, and let c>0c>0 be a constant small enough such that {z:∣f(z)∣≤c}\{ z: \lvert f(z)\rvert\leq c\} has mm distinct connected components.Source says disproved (Lean)Formalized
#1048If f∈C[x]f\in \mathbb{C}[x] is a monic polynomial with all roots satisfying ∣z∣≤r\lvert z\rvert \leq r for some r<2r<2, then must {z:∣f(z)∣<1}\{ z: \lvert f(z)\rvert <1\} have a connected component with diameter >2−r>2-r?Source says disproved (Lean)Formalized
#1049Let t>1t>1 be a rational number. Is ∑n=1∞1tn−1=∑n=1∞τ(n)tn\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n} irrational, where τ(n)\tau(n) counts the divisors of nn?Source says openFormalized
#1050No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1051Is it true that if a0<a1<a2<⋯a_0 < a_1 < a_2 < \cdots is a strictly increasing sequence of integers with lim inf⁡an1/2n>1\liminf a_n^{1/2^n} > 1, then the series ∑n=0∞1an⋅an+1\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}} is irrational?Source says proved (Lean)Formalized
#1052Are there only finitely many unitary perfect numbers?Source says openFormalized
#1053No statement retained — open to read what the source holdsSource says openNo formal declaration
#1054Let f(n)f(n) be the minimal integer mm such that nn is the sum of the kk smallest divisors of mm for some k≥1k\geq 1. Show that ff is undefined at n=2n=2, i.e. we get the junk value 00.Source says openFormalized
#1055A prime pp is in class 11 if the only prime divisors of p+1p+1 are 22 or 33. In general, a prime pp is in class rr if every prime factor of p+1p+1 is in some class ≤r−1\leq r-1, with equality for at least one prime factor. Are there infinitely many primes in each class?Source says openFormalized
#1056Let k≥2k ≥ 2. Does there exist a prime pp and consecutive intervals I0,…,IkI_0,\dots,I_k such that ∏n∈Iin≡1mod  n\prod\limits_{n{\in}I_i}n \equiv 1 \mod n for all 1≤i≤k1 \le i \le k?Source says openFormalized