Disjoint color-avoiding triangles

A set of pairwise edge-disjoint triangles of an edge-colored K_n is r-color avoiding if it does not contain r monochromatic triangles, each having a different color. Let f_r(n) be themaximum integer so that in every edge coloring of K_n with r colors, there is a set of f _r(n) pairwise edge-disjoint triangles that is r-color avoiding. We prove that 0.1177n²(1 - o(1)) < f2(n) < 0.1424n²(1 + o(1)). The proof of the lower bound uses probabilistic arguments, fractional relaxation and some packing theorems. We also prove that f_r(n)/n ² < 1/6 (1 - 0.145^r-1) + o(1). In particular, for every r, if n is sufficiently large, there are edge colorings of K_n with r colors so that the removal of any o(n²) members from any Steiner triple system does not turn it r-color avoiding.