Open-problem statements, with their sources attached.

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. The consistency of this was proved by Hajnal and Komjáth.

Determine which countable ordinals $β$ have the property that, if $α = \omega^β$, then in any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.

Erdős Problem 598:* Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$. Can one colour the countable subsets of $m$ using $\kappa$ many colours so that every $X \subseteq m$ with $|X| = \kappa$ contains subsets of all possible colours?

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. Does there exist a 2-colouring f : α → Fin 2 such that no A_i is monochromatic?

This is an open question about Property B for almost-disjoint families with a forbidden intersection size of 1. Note:* This generalises the formulation in which the ground set is ℕ. Since every countably infinite set is in bijection with ℕ, the two formulations are equivalent, but working over an arbitrary ground type makes the statement apply immediately to, e.g., almost-disjoint families of countable subsets of an uncountable space.

Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\not\in A$ for all $A$. Must there exist an infinite $Y\subseteq X$ that is independent - that is, for all finite $B\subset Y$ we have $f(B)\not\in Y$?

Erdős Problem 70*: Let $\mathfrak{c}$ be the cardinality of the continuum, let $\beta$ be a countable ordinal, and let $2 \le n < \omega$. Is it true that $\mathfrak{c} \to (\beta, n)^3_2$?

Note: The cases $n \le 3$ are trivially true (see omega_three), so the genuine content of the conjecture begins at $n = 4$.

First open case beyond Erdős–Rado*: $\mathfrak{c} \to (\omega \cdot 2, 4)^3_2$.

Erdős and Rado proved $\mathfrak{c} \to (\omega + n, 4)^3_2$ for every finite $n \ge 2$ (see erdos_rado), which covers all red ordinals below $\omega \cdot 2 = \omega + \omega$. This variant asks whether the result extends to $\beta = \omega \cdot 2$, the simplest countable ordinal not covered by their theorem.

*The relation at $\omega_1$**: $\mathfrak{c} \to (\omega_1, n)^3_2$ for finite $n \ge 2$, where $\omega_1 = \aleph_1$ is the first uncountable ordinal.

Note that $\omega_1$ is not a countable ordinal, so this is not directly an instance of the main Erdős problem (which asks for countable $\beta$). Under CH, $\omega_1 = \mathfrak{c}.\mathrm{ord}$, making this a self-referential question about $\mathfrak{c}.\mathrm{ord} \to (\mathfrak{c}.\mathrm{ord}, n)^3_2$.

Determine the value of the Busy Beaver function at n = 6.

The Vaught conjecture states that for a countable language L and a complete L-Theory T the number of countable models of T (up to isomorphism) is finite, $\aleph_0$ or $2^{\aleph_0}$.