51 problems

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

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

Csikvári–Gyarmati–Sárközy conjecture. The equation $a+b=c\cdot d$ is partition regular.

Gyarmati–Ruzsa conjecture. The equation $x^2+y^2=2z^2$ is partition regular.

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…

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

Let $a,b\in\mathbb N$ be positive integers, and consider finite colorings of $\mathbb N$. Sahasrabudhe's conjecture. The set … is not partition regular: there is a finite coloring…

Let $K$ be an imaginary quadratic field with ring of integers containing the notation above, and let $R_N$ denote the corresponding region of ring elements used in the quadratic av…

Let $A\mathbf{x}=0$ be a system of homogeneous equations whose solution set is partition-regular, and let $p_*(n)$ denote the threshold for $[n]_p$ to contain monochromatic solutio…

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

Quadratic Rado conjecture. The equation

Let a finite coloring of the natural numbers be a finite disjoint partition of $\mathbb{N}$, and call a configuration monochromatic if all its elements lie in one part of the parti…

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…

Nonlinear Rado conjecture. For every $p\in L$,

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…

Central-ultrafilter conjecture. The family of ultrafilters

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