8 problems

For a linear equation in $m$ variables, say that it is $k$-regular if every $k$-coloring of the positive integers yields a monochromatic solution, and say that it is regular if it…

Rado-number formula. Then

Let a linear homogeneous equation have nonzero integer coefficients, not all of the same sign. Golowich's conjecture. For each positive integer $k$ there is an integer $m(k)$ such…

Let $R_3(\mathcal{E})$ denote the smallest $n$ such that every 3-coloring of $[n]$ induces a monochromatic solution to the equation $\mathcal{E}$. For $m\ge 3$, consider the equati…

Let $\ell$ and $k$ be positive integers satisfying … For the 2-color Rado number $\operatorname{RR}(x+y+kz=\ell w)$, Modified Robertson–Myers conjecture. for fixed $\ell$, … The pa…

Let $cell$ and $k$ be positive integers satisfying … For the 2-color Rado number $\operatorname{RR}(x+y+kz=\ell w)$, Robertson–Myers conjecture. for fixed $\ell$, … This conjecture…

Let $a$ be a positive integer, and let $R(x+a(x+y)=2aw)$ denote the 2-color Rado number of the four-variable equation $x+a(x+y)=2aw$. Four-variable Rado number conjecture. … The co…

Let $q\in\mathbb{N}$, and let $R(x+qy=q^2z)$ denote the 2-color Rado number of the equation $x+qy=q^2z$. Parity conjecture. … The claim is presented as an empirical conjecture base…