  We provide multicolored and infinite generalizations for a Ramsey-type
problem raised by Bollob\'as, concerning colorings of $K_n$ where each color is
well-represented. Let $\chi$ be a coloring of the edges of a complete graph on
$n$ vertices into $r$ colors. We call $\chi$ $\varepsilon$-balanced if all
color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$,
together with an $r$-coloring of its edges. Consider the smallest natural
number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all
$\varepsilon$-balanced colorings $\chi$ of $K_n$ contain a subgraph isomorphic
to $H$ in its coloring. Bollob\'as conjectured a simple characterization of $H$
for which $R_\varepsilon^2(H)$ is finite, which was later proved by Cutler and
Mont\'agh. Here, we obtain a characterization for arbitrary values of $r$, as
well as asymptotically tight bounds. We also discuss generalizations to graphs
defined on perfect Polish spaces, where the corresponding notion of
balancedness is each color class being non-meagre.
