On colorful edge triples in edge-colored complete graphs

Abstract

An edge-coloring of the complete graph $K_n$ we call $F$-caring if it leaves no $F$-subgraph of $K_n$ monochromatic and at the same time every subset of $|V(F)|$ vertices contains in it at least one completely multicolored version of $F$. For the first two meaningful cases, when $F=K_{1,3}$ and $F=P_4$ we determine for infinitely many $n$ the minimum number of colors needed for an $F$-caring edge-coloring of $K_n$. An explicit family of $2\lceil {\log }_{2}n\rceil$ $3$-edge-colorings of $K_n$ so that every quadruple of its vertices contains a totally multicolored $P_4$ in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the five length cycle.

15 pages