Let $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$ be the set of subsets of $I$ of cardinality $m$. Let $f$ be a coloring of $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$ and $g$ a coloring of $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$. We write $f\to g$ if every $f$-homogeneous $H\subseteq I$ is also $g$-homogeneous. The least $m$ such that $f\to g$ for some $f:\left(\genfrac{}{}{0pt}{}{I}{m}\right)\to k$ is called the $k$-width of $g$ and denoted by $w_k\left(g\right)$. In the first part of the paper we prove the existence of colorings with high $k$-width. In particular, we show that for each $k>0$ and $m>0$ there is a coloring $g$ with $w_k\left(g\right)=m$. In the second part of the paper we give applications of wide colorings in the theory of generalized quantifiers....