5 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…

For $k\geq 2$, let $\sum_{<k}$ be the collection of words in $*_1,\ldots,*_d$ in which each letter occurs at most $k-1$ times. For elements $a,x_1,\ldots,x_d$ of an $s$-step nilpot…

Let $f^*(k,-1)$ be the least board size for which Maker can force a solution with distinct variables in the Maker–Breaker game for the equation $1/x_1+\cdots+1/x_k=1/y$, and let…

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…