Open-problem statements, with their sources attached.

[Ma21] showed that $C \leq 4$.

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. Is $h(n)=o(n)$?

If $\{1, \ldots, N\}$ is $r$-coloured then, for $N \geqslant N_0(r)$, there are integers $x, y \geqslant 3$ such that $x + y, xy$ have the same colour.

Find reasonable bounds for $N_0(r)$. The goal is to improve upon the Green-Sawhney bound.

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. How does $h(n)$ grow? Can we find a (good) explicit function $g$ such that $g = O(h)$ ?

If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set $\lbrace 0,1,3 \rbrace$ that $A$ can contain?

Conjectured in [Aa19] p.579: $\left({1}{3} + o(1)\right) n^2$.

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. How does $h(n)$ grow? Can we find a (good) explicit function $g$ such that $h = O(g)$ ?

Conjecture p.579 in [Aa19]: $\left({1}{3} + o(1)\right) n^2$.

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. How does $h(n)$ grow? Can we find a (good) explicit function $g$ such that $h = \Theta(g)$ ?

For which values of $k$ is the following true: whenever we partition $[N] = A_1 \cup \dots \cup A_k$, $\left|\bigcup^k_{i=1} (A_i \hat{+} A_i)\right| \geq \frac{1}{10} N$?

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. How does $h(n)$ grow? Can we find a (good) explicit function $g$ such that $g = o(h)$ ?

We conjecture that the best-known upper bound can be lowered.

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. How does $h(n)$ grow? Can we find a (good) explicit function $g$ such that $h = o(g)$ ?

We conjecture that the best-known lower bound can be raised.

Let $a_1< a_2 < ⋯$ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k \to \infty$.

The analogous problem in $\mathbb{F}^n_p$ remains open. [Gr24]

Let $a_1< a_2 < ⋯$ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k ^ {1 + c} \to 0$ for any $c > 0$.

What is the size of the smallest set $A \subset \mathbb{Z} / p\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation?

Suppose monotone sequence $A$ satisfies the following: A 0 = 1 and for all j, A (j + 1) is the smallest natural number that cannot be written as a sum of consecutive terms of A 0, ..., A j. Then it is conjectured that $a_k ~ \frac{k \log k}{\log \log k}$.

Propose a better lower bound along primes.

Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq \{1, \ldots, \lfloor c n \rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?