Disjoint Color-Avoiding Triangles

Abstract

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 the maximum 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 $0.1177n^2(1-o(1))<f_2(n)<0.1424n^2(1+o(1))$. The proof of the lower bound uses probabilistic arguments, fractional relaxation and some packing theorems. We also prove that $f_r(n)/n^2<\frac{1}{6}(1-0.145^{r-1})+o(1)$. 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 $o(n^2)$ members from any Steiner triple system does not turn it r-color avoiding.