  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 Gr\"otzsch graph is strictly larger than that of
the five length cycle.
