14 problems

Arithmetic removal conjecture. If $A\subseteq[N]$ and the number of vectors $x\in A^m$ satisfying $Lx=0$ is $o(N^{m-r})$, then

Let $L=(L_1,\ldots,L_t)$ be a non-degenerate genus-one translation-invariant system of equations over $\mathbb{F}_q$, where each $L_j$ is a translation-invariant equation in $s$ va…

Let $\mathcal{L}: c_1x_1 + \ldots +c_k x_k = 0$ be a homogeneous linear equation with $k \ge 3$, where $c_1,\ldots,c_k \in \mathbb{Z} \setminus \{0\}$ satisfy … but there exists a…

Odd-length uncommonness conjecture. The equation is uncommon over the integers.

Let \boldsymbol{[?] be a $(2\times k)$-system, and let $s(\Psi)$ denote the length of its shortest equation, where the length of an equation is the number of nonzero coefficients…

Consider a linear equation … where $a_1,\dots,a_k$ are integers. The equation has genus one if … and $\sum_{i\in T}a_i\neq 0$ for every nonempty proper subset $T\subsetneq [k]$. Le…

Let $L$ be a $(2\times 5)$-system of linear equations, and let $s(L)$ denote the parameter used in the paper. Assume that $s(L)=4$. Non-Sidorenko conjecture. Every $(2\times 5)$-sy…

Uncommon $(2\times k)$-system conjecture. For even $k\geq 6$ and sufficiently large odd $q$, every $(2\times k)$-system $L$ satisfying

Let $a_1,\dots,a_k\in\mathbb{Z}$ and consider the equation … A canceling partition is a partition of the coefficients into pairs $\{a_i,a_j\}$ such that $a_i+a_j=0$. Call the equat…

Let $[n]=\{1,2,\dots,n\}$, let $a_1,\dots,a_k\in\mathbb{Z}$, and consider the equation … For a coloring $f:[n]\to\{-1,1\}$, call a solution $(x_1,\dots,x_k)\in[n]^k$ monochromatic…

Let $q$ be a prime power, let $n$ be a positive integer, and let $k\ge 2$. Consider the linear equation over $\mathbb{F}_q^n$ … The equation is common if, for every two-coloring of…

For a positive integer $r$, let an equation be $r$-regular when every coloring of the positive integers with $r$ colors has a monochromatic positive-integer solution. Lower-bound c…

Let $\mathcal{F}$ be a possibly infinite set of systems of induced equations, and let a Boolean function be $\mathcal{F}$-free when it contains no induced solution of any system in…

Green's conjecture. Every system of homogeneous linear equations $Mx=0$ has the removal property.