M-p-c set ($(m,p,c)$-set)

An $(m,p,c)$-set with generators $z_1,\dots ,z_m$ consists of all positive integers of the form

For every positive $m,p,c$, every finite coloring of the positive integers contains a monochromatic M-p-c set.

For every dimension $m$, every finite coloring of the positive integers contains $x_1,\dots ,x_m$ for which all nonempty sums of distinct $x_i$ have one color. This is the case $p=c=1$ of the monochromatic m-p-c set theorem.