Open-problem statements, with their sources attached.

Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?

Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number $\leq\aleph_0$?

Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?

Is it true that, for $k\geq 4$, $f_k(n) \gg \frac{n^{1-\frac{1}{k-1}}}{(\log n)^{c_k}}$ for some constant $c_k>0$?

Let $S \subseteq \mathbb{R}$ be a set containing no solutions to $a + b = c$. Must there be a set $A \subseteq \mathbb{R} \setminus S$ of cardinality continuum such that $A + A \subseteq \mathbb{R}\setminus S$?

Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $\frac n 3 + Ω(n)$, where $Ω(n) → ∞$ as $n → ∞$?

Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$. Are there at least $\alpha^{15} N^{10}$ tuples $(x_1, \dots, x_5, y_1, \dots, y_5) \in G^{10}$ such that $x_i + y_j \in A$ whenever $j \in \{i, i+1, i+2\}$?

Note: We interpret indices modulo 5.

Is $W(k, r)$ a polynomial in $r$, for fixed $k$?

We formulate this as asking if $W(k, r)$ has polynomial growth in $r$. We know it is not the case for $k = 3$ [Gr21, p.3].

It remains an interesting open problem to actually write down a colouring showing (say) $W(3, r) \ge 2r^2$ for some $r$. [Gr24]

$W(3, 20) \ge 389$ from [AKS14, Table 2].

$W(3, 21) \ge 416$ from [AKS14, Table 2].

$W(3, 22) \ge 464$ from [AKS14, Table 2].

$W(3, 23) \ge 516$ from [AKS14, Table 2].

$W(3, 24) \ge 593$ from [AKS14, Table 2].

$W(3, 25) \ge 656$ from [AKS14, Table 2].

$W(3, 26) \ge 727$ from [AKS14, Table 2].

$W(3, 27) \ge 770$ from [AKS14, Table 2].

$W(3, 28) \ge 827$ from [AKS14, Table 2].

$W(3, 29) \ge 868$ from [AKS14, Table 2].

$W(3, 30) \ge 903$ from [AKS14, Table 2].