Restricted generalized Schur numbers

Abstract

For $k\geq2$, let $S_r(k;\ell)$ be the smallest $n$, if exists, such that every $r$-coloring of $\{1,2,\ldots,n\}$ has a monochromatic solution $\mathcal{S}$ to the equation \[ x_1+x_2+\cdots+x_k=x_{k+1} \] such that the number of distinct integers in $\mathcal{S}$ is exactly $\ell+1$. We prove that, if $\ell\geq2$ is fixed, then \[ S_2(k;\ell)= k^2+\left[\frac{(\ell+1)(\ell-2)}{2}+2\right]k+\ell(\ell-2) \] for all large enough $k$. In particular, we have $S_2(k;2)=k^2+2k$ for all $k\geq3$.