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generalized quantifier; Ramsey theory
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 {\sl $k$-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 $t=(1,\ldots,1)$ 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.
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