We solve four similar problems: for every fixed
and large
, we describe all values
of
such that for every
-edge-coloring of the
complete
-partite
graph
there exists a
monochromatic (i) cycle
with
vertices,
(ii) cycle
with
at least
vertices,
(iii) path
with
vertices,
and (iv) path
with
vertices.
This implies a generalization for large
of the
conjecture by Gyárfás, Ruszinkó, Sárközy and Szemerédi that for every
-edge-coloring of the
complete
-partite
graph
there is a
monochromatic path
.
An important tool is our recent stability theorem on monochromatic connected
matchings.
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