Abstract:Let $\binom {I}{m}$ be the set of subsets of $I$ of cardinality $m$. Let $f$ be a coloring of $\binom {I}{m}$ and $g$ a coloring of $\binom {I}{m}$. We write $f\rightarrow g$ if every $f$-homogeneous $H\subseteq I$ is also $g$-homogeneous. The least $m$ such that $f\rightarrow g$ for some $f:\binom {I}{m}\rightarrow k$ is called the {$k$-width} of $g$ and denoted by $w_k(g)$. 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 $w_k(g)=m$. In the second part of the paper we give applications of wide colorings in the theory of generalized quantifiers. In particular, we show that for every monadic similarity type $t=(1,...,1)$ there is a generalized quantifier of type $t$ which is not definable in terms of a finite number of generalized quantifiers of a smaller type.
Keywords: generalized quantifier, Ramsey theory
AMS Subject Classification: 05C55, 03C