  Erd\H{o}s and Szekeres's quantitative version of Ramsey's theorem asserts
that any complete graph on n vertices that is edge-colored with two colors has
a monochromatic clique on at least 1/2log(n) vertices. The famous
Erd\H{o}s-Hajnal conjecture asserts that forbidding fixed color patterns
ensures larger monochromatic cliques. Specifically, it claims that for any
fixed integer k and any clique K on k vertices edge-colored with two colors,
there is a positive constant a such that in any complete n-vertex graph
edge-colored with two colors that does not contain a copy of K, there is a
monochromatic clique on at least n^a vertices.
  We consider edge-colorings with three colors. For a family H of triangles,
each colored with colors from {r, b, y}, Forb(n,H) denotes a family of
edge-colorings of the complete n-vertex graph using colors from {r, b, y} and
containing none of the colorings from H. Let h_2(n, H) be the maximum q such
that any coloring from Forb(n, H) has a clique on at least q vertices using at
most two colors. We provide bounds on h_2(n, H) for all families H consisting
of at most three triangles. For most of them, our bounds are asymptotically
tight. This extends a result of Fox, Grinshpun, and Pach, who determined h_2(n,
H) for H consisting of a rainbow triangle, and confirms the multicolor
Erd\H{o}s-Hajnal conjecture for these sets of patterns.
