Disjoint Color-Avoiding Triangles

A set of pairwise edge-disjoint triangles of an edge-colored Kn is r-color avoiding if it does not contain r monochromatic triangles, each having a different color. Let fr(n) be the maximum integer so that in every edge coloring of Kn with r colors, there is a set of fr(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 fr(n)/n < 1 6 (1 − 0.145r−1) + o(1). In particular, for every r, if n is sufficiently large, there are edge colorings of Kn with r colors so that the removal of any o(n2) members from any Steiner triple system does not turn it r-color avoiding.