29 problems

Ergodic Pythagorean recurrence conjecture. There exists a solution $(x,y,z)\in\mathbb{N}^3$ of $x^2+y^2=z^2$ such that

Generalized Pythagorean partition-regularity conjecture. The equation

Erdős–Graham conjecture. Any finite colouring of $\mathbb{N}$ has a monochromatic Pythagorean triple.

Ergodic variant of the quadratic Rado conjecture. There exist distinct $x,y,z\in\mathbb{N}$ such that

Quadratic Rado conjecture. The equation

Density-regularity characterization conjecture. There exists a computable collection $\{B_n\}_{n=1}^{\infty}$ of subsets of $R\setminus\{0\}$ such that

Partition-regularity characterization conjecture. There exists such a computable collection of finite partitions $\{\mathcal{C}_n\}_{n=1}^{\infty}$, with…

Nonexistence conjecture. For any $n\in\mathbb{N}_{>1}$, $\star_n$-idempotent ultrafilters do not exist.

Generalized Pythagorean-pair conjecture. If at least one of $ac$, $bc$, or $(a+b)c$ is a square, then $ax^2+by^2=cz^2$ is partition regular with respect to $x,y$.

Rado triple conjecture. The equation

FKM's pairwise partition-regularity conjecture. If at least one of $ac$, $bc$, or $(a+b)c$ is a square, then $a x^2+b y^2=c z^2$ is partition regular with respect to $x,y$.

For $k\in\mathbb{N}$, let $a_1,\ldots,a_k\in\mathbb{Z}/(2^r\mathbb{Z})$, and let $d\leq r$ be the largest non-negative integer such that $2^d$ divides $\sum_{i\in I}a_i$ for some n…

Alweiss's non-partition-regularity conjecture. There exists $n$ such that, for every $d\in\mathbb{N}$, the pattern

Three-term reciprocal-pattern conjecture. The pattern

Reciprocal parallelogram conjecture. The pattern

Let $T$ denote the polynomial setting used in the paper, and let $\mathbb{N}$ denote the positive integers. A polynomial pattern $\mathcal{F}$ is partition regular over a domain if…

For $d\geq 1$, let $P_d$ be the set of polynomials in countably many variables with nonnegative integer coefficients, constant term, and degree at most $d$ in each variable. A patt…

For $d\geq 1$, let $P_d$ be the set of polynomials in countably many variables with nonnegative integer coefficients, constant term, and degree at most $d$ in each variable. A fini…

An equation is partition regular if every finite coloring of $\mathbb{N}$ admits a solution whose variables all have the same color. Erdős–Graham conjecture. For every finite color…

Partition-regularity conjecture. For every choice of non-zero integers $a,b,c$, this equation is partition regular.

Let $P\in \mathbb{Z}[x_{1},\ldots,x_{n}]$ be a polynomial, and define its corresponding monovariate polynomial by … Assume that $\widetilde{P}$ is a nonzero homogeneous linear poly…

Let $\mathbb{Q}\setminus\{0\}$ be partitioned into finitely many sets, … For a sequence $\langle x_n\rangle_{n=1}^{\infty}$, let $FS(\langle x_n\rangle_{n=1}^{\infty})$ denote its…

Let $G=B_1\cup\dotsm\cup B_q$ be a finite partition of a semimodule $(G,\pmb{+})$ over a semiring $(R,+,\cdot)$. For a subset $F\subseteq R$, write $Fb=\{fb:f\in F\}$. Schur–Brauer…

Image maximality conjecture. The system ${\bf D}$ is image maximal.

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…