35 problems

Rado's conjecture. For every $n\in\mathbb{Z}^+$, there exists a linear homogeneous equation over $\mathbb{Z}$ with degree of regularity equal to $n$.

Fox–Radoičić's conjecture. This equation has degree of regularity $n-1$. The conjecture was introduced as a simpler family that would establish Rado's 1933 conjecture; this paper p…

Growth-rate conjecture. The correct order of magnitude in $r$ for $\widetilde A(n,r)$ should be of the shape

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…

Let $a_1x_1+\dots+a_kx_k=0$ be a linear equation with nonzero integer coefficients. The equation is $2$-common over the integers if every $2$-coloring of $[n]=\{1,\dots,n\}$ has as…

Optimal-coloring conjecture. The coloring that colors multiples of $4$ red and all other integers blue is optimal for the equation $4x+4y=z$.

Costello–Elvin conjecture. If $\mathcal{E}$ is a three-variable equation of the form $ax+by=cz$ with $a,b,c\in\mathbb{N}$, then

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

Costello–Elvin's conjecture. Every 2-coloring of $[n]$ should have at least

Costello–Elvin's conjecture. The equation is common over the integers if and only if $k$ is even and $a_1,\ldots,a_k$ can be partitioned into pairs whose sums are zero.

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…

Consider a system of linear forms whose image has codimension two and which comprises an even number of linear forms. Kamčev–Liebenau–Morrison conjecture. An analogue of the lemma…

Let $k$ be a positive integer and let $(a_1,a_2,\ldots,a_k)$ be a sequence of non-zero integers. Consider the infinite system … The system is partition regular in $\mathbb N$ if ev…

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…

Let $q$ be a prime power, and let $L=a_1x_1+\dots+a_kx_k$ be a single linear form over $\mathbbm{F}_q$, with each coefficient $a_i\in\mathbbm{F}_q^\times$. The coefficients satisfy…

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

Fox–Pham–Zhao conjecture. The equation is common only if its coefficients can be partitioned into pairs, with the two coefficients in each pair summing to zero.

Consider a linear equation whose coefficients can be partitioned into pairs such that the coefficients in each pair sum to zero. For an equation with an even number $d$ of variable…

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…

Kucharz's conjecture. If the equation admits a continuous solution $(\varphi_1,\dots,\varphi_r)$, then it admits a Nash regulous solution.