Scientists / Open questions

保罗·埃尔德什

Paul Erdős

数论、组合数学、图论

41 个问题

Exact determination of the Erdős–Reid–Schelp–Staton function $q(n,j,l)$ : the minimum size of a graph in which every $n+j$ vertices induce a subgraph of maximum degree at least $n$
The unresolved measure-theoretic questions from Erdős's 1953 paper 'Some remarks on set theory. IV': Theorem 5 for mixed families of null and first-category sets, and positive-measure families with empty $\aleph_1$ -subfamily intersections
The unresolved regime of the Erdős–Gyárfás function $f(n,p,q)$ : superlinearity between the linear and quadratic thresholds, threshold constants, and geometric variants
The Erdős–Fishburn total distance-count problem: the asymptotics of $\Sigma_n$ and the triangular-lattice structure of its minimizers
Open problems of Erdős and Fishburn on the fine structure of the planar few-distance extremal functions $g(k)$ and $f(n)$
The leading constant in the Bollobás–Erdős blow-up function $c_r(ε)$ as $ε$ tends to 0
The general hypergraph Turán problem: determining $\operatorname{ex}(n,\mathscr{L})$ and extremal structures for forbidden families of uniform hypergraphs
Free sets for set mappings: the open boundary cases of the Erdős–Hajnal free-set problems of 1958
Characterizing the sequences of rationals realized by partitions of the unit fractions into finite subsets (Erdős–Stein, 1963)
The Bollobás–Erdős triangle degree-sum problem at the Mantel threshold: determine the guaranteed maximum of d(x)+d(y)+d(z) over triangles in graphs with ⌊n²/4⌋+1 edges
Linear-size crossing families in planar point sets (the crossing family problem)
The logarithmic gap in the Erdős–Gyárfás–Łuczak function $f_r(k)$ : is $f_r(k)=\Theta_r(k^{r/(r-1)})$ for fixed $r\ge 3$ ?
Exact value of the supremum $R$ of reciprocal sums of sum-free sequences (Erdős, 1962)
The optimal symbol-multiplicity threshold guaranteeing a full Latin transversal in an $n\times n$ array
Local constraints ensuring small representing sets: the local-to-global covering relation $(p,s)\underset{r}{\longrightarrow}t$ of Erdős–Hajnal–Tuza
Exact small values of the Erdős–Fishburn equidistant-neighbor function: uniqueness of the extremal 8-point configuration and the value of g(5)
Schlicht Taylor series converging uniformly but not absolutely on the unit circle: must the exponents satisfy more than $\sum 1/m_k=\infty$ (Erdős–Herzog–Piranian, 1951)?
Determining the density–size tradeoff $c(k,\varepsilon)$ for forcing small subgraphs of minimum degree $k$
Exponential growth rate of the minimum order of a complete graph decomposable into factors of given star number
Sharp with-high-probability asymptotics of clique coverings of the random graph $G(n,p)$ for constant edge probability
Independent sets for small pictures of the reals and measure/category dichotomies for graphs on the reals: the surviving open questions of Erdős–Hajnal, Some remarks on set theory. VIII (1960)
Sets of divergence of sequences of linear fractional transformations: the general classification and closure under finite intersections
Open threads from Erdős–Hajnal–Milner's almost-disjoint transversal theory: the $\mathscr S \Rightarrow \mathscr H$ converse, necessity at inaccessible cardinals, and removal of cardinal-arithmetic hypotheses
The regime $c<2/5$ of the edge-minimum function $K(c)$ for maximal triangle-free graphs with bounded maximum degree

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Exact determination of the Erdős–Reid–Schelp–Staton function q(n,j,l)q(n,j,l)q(n,j,l): the minimum size of a graph in which every n+jn+jn+j vertices induce a subgraph of maximum degree at least nnn

The unresolved measure-theoretic questions from Erdős's 1953 paper 'Some remarks on set theory. IV': Theorem 5 for mixed families of null and first-category sets, and positive-measure families with empty ℵ1\aleph_1ℵ1​-subfamily intersections

The unresolved regime of the Erdős–Gyárfás function f(n,p,q)f(n,p,q)f(n,p,q): superlinearity between the linear and quadratic thresholds, threshold constants, and geometric variants

The Erdős–Fishburn total distance-count problem: the asymptotics of Σn\Sigma_nΣn​ and the triangular-lattice structure of its minimizers

Open problems of Erdős and Fishburn on the fine structure of the planar few-distance extremal functions g(k)g(k)g(k) and f(n)f(n)f(n)

The leading constant in the Bollobás–Erdős blow-up function cr(ε)c_r(ε)cr​(ε) as εεε tends to 0

The general hypergraph Turán problem: determining ex⁡(n,L)\operatorname{ex}(n,\mathscr{L})ex(n,L) and extremal structures for forbidden families of uniform hypergraphs

Free sets for set mappings: the open boundary cases of the Erdős–Hajnal free-set problems of 1958

Characterizing the sequences of rationals realized by partitions of the unit fractions into finite subsets (Erdős–Stein, 1963)

The Bollobás–Erdős triangle degree-sum problem at the Mantel threshold: determine the guaranteed maximum of d(x)+d(y)+d(z) over triangles in graphs with ⌊n²/4⌋+1 edges

Linear-size crossing families in planar point sets (the crossing family problem)

The logarithmic gap in the Erdős–Gyárfás–Łuczak function fr(k)f_r(k)fr​(k): is fr(k)=Θr(kr/(r−1))f_r(k)=\Theta_r(k^{r/(r-1)})fr​(k)=Θr​(kr/(r−1)) for fixed r≥3r\ge 3r≥3?

Exact value of the supremum RRR of reciprocal sums of sum-free sequences (Erdős, 1962)

The optimal symbol-multiplicity threshold guaranteeing a full Latin transversal in an n×nn\times nn×n array

Local constraints ensuring small representing sets: the local-to-global covering relation (p,s)⟶rt(p,s)\underset{r}{\longrightarrow}t(p,s)r⟶​t of Erdős–Hajnal–Tuza

Exact small values of the Erdős–Fishburn equidistant-neighbor function: uniqueness of the extremal 8-point configuration and the value of g(5)

Schlicht Taylor series converging uniformly but not absolutely on the unit circle: must the exponents satisfy more than ∑1/mk=∞\sum 1/m_k=\infty∑1/mk​=∞ (Erdős–Herzog–Piranian, 1951)?

Determining the density–size tradeoff c(k,ε)c(k,\varepsilon)c(k,ε) for forcing small subgraphs of minimum degree kkk

Exponential growth rate of the minimum order of a complete graph decomposable into factors of given star number

Sharp with-high-probability asymptotics of clique coverings of the random graph G(n,p)G(n,p)G(n,p) for constant edge probability

Independent sets for small pictures of the reals and measure/category dichotomies for graphs on the reals: the surviving open questions of Erdős–Hajnal, Some remarks on set theory. VIII (1960)

Sets of divergence of sequences of linear fractional transformations: the general classification and closure under finite intersections

Open threads from Erdős–Hajnal–Milner's almost-disjoint transversal theory: the S⇒H\mathscr S \Rightarrow \mathscr HS⇒H converse, necessity at inaccessible cardinals, and removal of cardinal-arithmetic hypotheses

The regime c<2/5c<2/5c<2/5 of the edge-minimum function K(c)K(c)K(c) for maximal triangle-free graphs with bounded maximum degree

Exact determination of the Erdős–Reid–Schelp–Staton function $q(n,j,l)$ : the minimum size of a graph in which every $n+j$ vertices induce a subgraph of maximum degree at least $n$

The unresolved measure-theoretic questions from Erdős's 1953 paper 'Some remarks on set theory. IV': Theorem 5 for mixed families of null and first-category sets, and positive-measure families with empty $\aleph_1$ -subfamily intersections

The unresolved regime of the Erdős–Gyárfás function $f(n,p,q)$ : superlinearity between the linear and quadratic thresholds, threshold constants, and geometric variants

The Erdős–Fishburn total distance-count problem: the asymptotics of $\Sigma_n$ and the triangular-lattice structure of its minimizers

Open problems of Erdős and Fishburn on the fine structure of the planar few-distance extremal functions $g(k)$ and $f(n)$

The leading constant in the Bollobás–Erdős blow-up function $c_r(ε)$ as $ε$ tends to 0

The general hypergraph Turán problem: determining $\operatorname{ex}(n,\mathscr{L})$ and extremal structures for forbidden families of uniform hypergraphs

The logarithmic gap in the Erdős–Gyárfás–Łuczak function $f_r(k)$ : is $f_r(k)=\Theta_r(k^{r/(r-1)})$ for fixed $r\ge 3$ ?

Exact value of the supremum $R$ of reciprocal sums of sum-free sequences (Erdős, 1962)

The optimal symbol-multiplicity threshold guaranteeing a full Latin transversal in an $n\times n$ array

Local constraints ensuring small representing sets: the local-to-global covering relation $(p,s)\underset{r}{\longrightarrow}t$ of Erdős–Hajnal–Tuza

Schlicht Taylor series converging uniformly but not absolutely on the unit circle: must the exponents satisfy more than $\sum 1/m_k=\infty$ (Erdős–Herzog–Piranian, 1951)?

Determining the density–size tradeoff $c(k,\varepsilon)$ for forcing small subgraphs of minimum degree $k$

Sharp with-high-probability asymptotics of clique coverings of the random graph $G(n,p)$ for constant edge probability

Open threads from Erdős–Hajnal–Milner's almost-disjoint transversal theory: the $\mathscr S \Rightarrow \mathscr H$ converse, necessity at inaccessible cardinals, and removal of cardinal-arithmetic hypotheses

The regime $c<2/5$ of the edge-minimum function $K(c)$ for maximal triangle-free graphs with bounded maximum degree