The two coloring of the edges of
K_{39} which establishes the lower bound is
available as a 0-1
adjacency matrix
in which color 0 is K_{3}-free and color 1
is K_{10}-free.
It was constructed by starting with the unique
(3,9) coloring on 35 vertices, adding 4 vertices
whose edges were colored randomly, and applying
simulated annealing.
This may not be the same graph that I published
some years ago in SIAM Discrete Math.

Geoff Exoo