Questions on the subject additive combinatorics, also known as arithmetic combinatorics, such as questions on: additive bases, sum sets, inverse sum set theorems, sets with small doubling, Sidon sets, Szemerédi's theorem and its ramifications, Gowers uniformity norms, etc. Often combined with the top-level tags nt.number-theory or co.combinatorics. Some additional tags are available for further specialization, including the tags sumsets and sidon-sets.

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### Density version of the Erdos-Graham conjecture

In 2003 E. S. Croot [Ann. of Math. 157(2)(2003), 545-556] proved the Erdos-Graham Conjecture which states that if $\{2,3,\ldots\}$ is partitioned into ﬁnitely many subsets then one of the subsets ...
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### Is $AA+A$ always at least as large as $A/A$?

Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A/A|$? In the line above, $AA+A:=\{ab+c:a,b,c \in A \}$, while $A/A:=\{a/b:a,b \in A, b\neq 0 \}$ is the ratio set. ...
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### Does every $n\times n\times n$ Latin cube contain a Latin transversal?

In 1967 H. J. Ryser conjectured that every Latin square of odd order has a Latin transversal. Similar to Latin squares, we may consider Latin cubes. QUESTION: Let $n$ be any positive integer. Does ...
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### Maximum-size sum sets

Let $A, B \subseteq G$ for abelian group $G$. It is obvious that $|A+B| \leq |A| \cdot |B|$. Is there an easy characterization of sets $A, B$ meeting this bound? I am especially interested in the case ...
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### Balancing points with lines

$\newcommand{\F}{\mathbb F}$ Suppose that $p$ is a prime, and $k<p/2$ a positive integer. Consider a system of $k$ distinct directions in the affine plane $\F_p^2$, and the system of $k$ pencils ...
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### Is every integer $n>1$ the sum of two squares and two central binomial coefficients?

Those integers $\binom{2n}n\ (n=0,1,2,\ldots)$ are called central binomial coefficients. By Stirling's formula, $$\binom{2n}n\sim \frac{4^n}{\sqrt{n\pi}}\ \ \ \ (n\to+\infty).$$ Of course, the ...
### Is every integer $n>1$ the sum of two triangular numbers and two powers of $5$?
Recall that the triangular numbers are those integers $$T_n=n(n+1)/2\ \ \ (n=0,1,2,\ldots).$$ In 1796 Gauss proved that each $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three triangular numbers, ...
### Decomposing a subset of $\mathbf Z$ into a sumset of irreducibles
We say that a subset $A$ of $\mathbf Z$ is irreducible if $|A| \ge 2$ and there do not exist $X, Y \subseteq \mathbf Z$ with $|X|, |Y| \ge 2$ such that $A = X + Y$. If $X \subseteq \mathbf Z$, we ...