Some results in polychromatic Ramsey theory

Abstract

Classical Ramsey theory(at least in its simplest form) is concerned with problems of the following kind: given a setXand a colouring of the set[X]nof unorderedn-tuples fromX, find a subsetY ⊆ Xsuch that all elements of[Y]nget the same colour. Subsets with this property are calledmonochromaticorhomogeneous, and a typical positive result in Ramsey theory has the form that whenXis large enough and the number of colours is small enough we can expect to find reasonably large monochromatic sets.Polychromatic Ramsey theoryis concerned with a “dual” problem, in which we are given a colouring of[X]nand are looking for subsetsY ⊆ Xsuch that any two distinct elements of[Y]ngetdifferentcolours. Subsets with this property are calledpolychromaticorrainbow. Naturally if we are looking for rainbow subsets then our task becomes easier when there are many colours. In particular given an integerkwe say that a colouring isk-boundedwhen each colour is used for at mostkmanyn-tuples.At this point it will be convenient to introduce a compact notation for stating results in polychromatic Ramsey theory. We recall that in classical Ramsey theory we writeto mean “every colouring of[κ]ninkcolours has a monochromatic set of order type α”. We will writeto mean “everyk-bounded colouring of[κ]nhas a polychromatic set of order type α”. We note that whenκis infinite andkis finite ak-bounded colouring will use exactlyκcolours, so we may as well assume thatκis the set of colours used.