Erdős Problem #1068
Does every graph with chromatic number aleph_1 contain a countable subgraph which is infinitely connected?
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Open problems in infinite combinatorics and set theory
Partition relations for infinite cardinals and ordinals, many of them from Erdős and his collaborators. Some are known to depend on axioms beyond ZFC.
Erdős Problem #1167
Erdős Problem 1167. Let r ≥ 2 be finite, γ ≥ 2, and λ be an infinite cardinal. Let κ_α > r be cardinals for all α < γ. Is it true that 2^λ → (κ_α + 1)_α < γ^r+1 implies λ → (κ_α)_α < γ^r? Here + means cardinal addition, so that κ_α + 1 = κ_α if κ_α is infinite. A problem of Erdős, Hajnal, and Rado.
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Erdős Problem #1175
Let κ be an uncountable cardinal. Must there exist a cardinal λ such that every graph with chromatic number λ contains a triangle-free subgraph with chromatic number κ? Shelah proved that a negative answer is consistent when κ = λ = aleph_1 (see erdos_1175.variants.aleph_one).
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Erdős Problem #1176
Let G be a graph with chromatic number aleph_1. Is it true that there is a colouring of the edges with aleph_1 many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours? A problem of Erdős, Galvin, and Hajnal.
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Erdős Problem #501
For every x ∈ ℝ let A_x ⊂ ℝ be a bounded set with outer measure < 1. Must there exist an infinite independent set, that is, some infinite X ⊆ ℝ such that x ∉ A_y for all x ≠ y ∈ X? If the sets A_x are closed and have measure < 1, then must there exist an independent set of size 3?
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Erdős Problem #592
Determine which countable ordinals β have the property that, if α = ω^β, then in any red/blue colouring of the edges of K_α there is either a red K_α or a blue K_3.
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Erdős Problem #593
Erdős Problem 593 (\500): Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number > aleph_0. The answer is the set of obligatory finite 3-uniform hypergraphs, represented here on the labelled vertex sets Fin n.
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Erdős Problem #595
Erdős Problem 595 (\250): Is there an infinite graph G which contains no K_4 and is not the union of countably many triangle-free graphs? A problem of Erdős and Hajnal [Er87].
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Erdős Problem #596
Erdős Problem 596 (Erdős–Hajnal, [Er87]). For which graph pairs (G_1, G_2) is it true that (1) for every n ≥ 1 there is a graph H without a G_1 such that any n-colouring of H's edges contains a monochromatic G_2, and yet (2) for every graph H without a G_1 there is an aleph_0-colouring of H's edges…
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Erdős Problem #598
Erdős Problem 598: Let m be an infinite cardinal and κ be the successor cardinal of 2^aleph_0. Can one colour the countable subsets of m using κ many colours so that every X ⊆ m with |X| = κ contains subsets of all possible colours?
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Erdős Problem #602
Does every almost-disjoint family of countably infinite sets whose pairwise intersections all have size ≠ 1 have Property B? Formally: let α be any type, let (A_i)_i ∈ I be a family of countably infinite subsets of α such that for all i ≠ j, the intersection A_i ∩ A_j is finite and |A_i ∩ A_j| ≠ 1.
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Erdős Problem #623
Let X be a set of cardinality aleph_ω and f be a function from the finite subsets of X to X such that f(A)not∈ A for all A. Must there exist an infinite Y⊆ X that is independent - that is, for all finite B⊂ Y we have f(B)not∈ Y?
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Erdős Problem #70
Erdős Problem 70: Let c be the order type of the real numbers, let β be a countable ordinal, and let 2 ≤ n < ω. Is it true that c → (β, n)^3_2? Note: The cases n ≤ 3 are trivially true (compare omega_three), so the genuine content of the conjecture begins at n = 4.
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