Open-problem statements, with their sources attached.

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].

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. Find the simplest $g(N)$ such that $R(N) = O(g(N))$.

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

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. Find the simplest $g(N)$ such that $R(N) = o(g(N))$.

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

Is it true that $f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}$ for all $\epsilon>0$?

This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'.

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

Is it true that if $m < k$ then $f_{k,m}(x) \gg x^{m/k}$ for sufficiently large $x$?

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

For $k>2$ it is not known if $f_{k,k}(x)=o(x)$.

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

Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct?

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

Probably $f(x) = x^5$ has the property that the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct.

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

Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it true that $f_{k, 3}(x) \gg x ^ (3 / k)$?

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

Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it even true that $f_{k, 3}(x) \gg_{\epsilon} x ^ (3 / k - \epsilon)$?

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

Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a for some a in A and n ≥ 0. What is the smallest possible value of lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2)?