  We solve four similar problems: For every fixed $s$ and large $n$, we
describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring
of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a
monochromatic (i) cycle $C_{2n}$ with $2n$ vertices, (ii) cycle $C_{\geq 2n}$
with at least $2n$ vertices, (iii) path $P_{2n}$ with $2n$ vertices, and (iv)
path $P_{2n+1}$ with $2n+1$ vertices.
  This implies a generalization for large $n$ of the conjecture by Gy\'arf\'as,
Ruszink\'o, S\'ark\H{o}zy and Szemer\'edi that for every $2$-edge-coloring of
the complete $3$-partite graph $K_{n,n,n}$ there is a monochromatic path
$P_{2n+1}$. An important tool is our recent stability theorem on monochromatic
connected matchings.
