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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
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Problems

35 Problems

NumberQuestionOpen
#70Erdős Problem 70: Let c\mathfrak{c} be the cardinality of the continuum, let β\beta be a countable ordinal, and let 2≤n<ω2 \le n < \omega. Is it true that c→(β,n)23\mathfrak{c} \to (\beta, n)^3_2?openFormalized
#111No statement retained — open to read what the source holdsopenNo formal declaration
#118No statement retained — open to read what the source holdsdisprovedNo formal declaration
#474No statement retained — open to read what the source holdsnot provableNo formal declaration
#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
#590Let αα be the infinite ordinal ωω\omega^{\omega}. It was proved by Chang [Ch72] that any red/blue colouring of the edges of KαK_α there is either a red KαK_α or a blue K3K_3.provedFormalized
#591Let αα be the infinite ordinal ωω2\omega^{\omega^2}. Is it true that any red/blue colouring of the edges of KαK_α there is either a red KαK_α or a blue K3K_3?provedFormalized
#592Determine which countable ordinals ββ have the property that, if α=ωβα = \omega^β, then in any red/blue colouring of the edges of KαK_α there is either a red KαK_α or a blue K3K_3.openFormalized
#593Erdős Problem 593 (500): Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number >ℵ0> \aleph_0.openFormalized
#594Erdős Problem 594 (Erdős–Hajnal [ErHa66], [Er69b]):provedFormalized
#595Erdős Problem 595 (250): Is there an infinite graph G which contains no K4K_4 and is not the union of countably many triangle-free graphs?openFormalized
#596Erdős Problem 596 (Erdős–Hajnal, [Er87]). For which graph pairs (G1,G2)(G_1, G_2) is it true thatopenFormalized
#597No statement retained — open to read what the source holdsopenNo formal declaration
#598Erdős Problem 598: Let mm be an infinite cardinal and κ\kappa be the successor cardinal of 2ℵ02^{\aleph_0}. Can one colour the countable subsets of mm using κ\kappa many colours so that every X⊆mX \subseteq m with ∣X∣=κ|X| = \kappa contains subsets of all possible colours?openFormalized
#599Erdős Problem 599 (the Erdős–Menger conjecture).provedFormalized
#601No statement retained — open to read what the source holdsopenNo formal declaration
#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
#623Let XX be a set of cardinality ℵω\aleph_\omega and ff be a function from the finite subsets of XX to XX such that f(A)∉Af(A)\not\in A for all AA. Must there exist an infinite Y⊆XY\subseteq X that is independent - that is, for all finite B⊂YB\subset Y we have f(B)∉Yf(B)\not\in Y?openFormalized
#1067Does every graph with chromatic number ℵ1\aleph_1 contain an infinitely connected subgraph with chromatic number ℵ1\aleph_1?disproved (Lean)Formalized
#1068Does every graph with chromatic number ℵ1\aleph_1 contain a countable subgraph which is infinitely connected?openFormalized
#1119Let m\mathfrak{m} be an infinite cardinal with ℵ0<m<c=2ℵ0\aleph_0 < \mathfrak{m} < \mathfrak{c} = 2^{\aleph_0}. Let {fα}\{f_\alpha\} be a family of entire functions such that, for every z0∈Cz_0 \in \mathbb{C}, there are at most m\mathfrak{m} distinct values of fα(z0)f_\alpha(z_0). Must {fα}\{f_\alpha\} have cardinality at most m\mathfrak{m}?independentFormalized
#1127No statement retained — open to read what the source holdsindependentNo formal declaration
#1128Erdős Problem 1128 (disproved by Prikry–Mills, 1978):disprovedFormalized
#1167The partition relation μ→(ν)1r\mu \to (\nu)^r_1 with a single color is equivalent to ν≤μ\nu \le \mu.openFormalized
#1168No statement retained — open to read what the source holdsopenNo formal declaration
#1169No statement retained — open to read what the source holdsnot disprovableNo formal declaration
#1170No statement retained — open to read what the source holdsopenNo formal declaration
#1171No statement retained — open to read what the source holdsopenNo formal declaration
#1172No statement retained — open to read what the source holdsopenNo formal declaration
#1173No statement retained — open to read what the source holdsopenNo formal declaration
#1174No statement retained — open to read what the source holdsnot disprovableNo formal declaration
#1175Let κ\kappa be an uncountable cardinal. Must there exist a cardinal λ\lambda such that every graph with chromatic number λ\lambda contains a triangle-free subgraph with chromatic number κ\kappa?openFormalized
#1176Let GG be a graph with chromatic number ℵ1\aleph_1. Is it true that there is a colouring of the edges with ℵ1\aleph_1 many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours?not disprovableFormalized
#1177No statement retained — open to read what the source holdsopenNo formal declaration

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