Rado's boundedness conjecture

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 is $k$-regular for every positive integer $k$. Rado's boundedness conjecture. For all $m\ge1$, there is a number $\Delta(m)$ such that if a linear equation in $m$ variables is $\Delta(m)$-regular, then it is regular.

The conjecture asks whether regularity can be detected using a number of colors depending only on the number of variables. It is known to hold if it holds for homogeneous equations, and the first nontrivial homogeneous three-variable case is established with the bound $24$; the optimality of that bound and the general conjecture remain open.