Combinatorics and quantifiers

Let I m be the set of subsets of I of cardinality m. Let f be a coloring of I m and g a coloring of I m . We write f → g if every f -homogeneous H ⊆ I is also g-homogeneous. The least m such that f → g for some f : I m → k is called the kwidth of g and denoted by wk(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 wk(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.