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Some results in polychromatic Ramsey theory

Published online by Cambridge University Press:  12 March 2014

Uri Abraham ,
James Cummings  and
Clifford Smyth
Uri Abraham
Affiliation:
Mathematics and Computer Science Department, Ben-Gurion University, Beer-Sheva, 84105, Israel. E-mail: abraham@math.bgu.ac.il
James Cummings
Affiliation:
Mathematical Sciences Department, Carnegie Mellon University, Pittsburgh PA 15215, USA. E-mail: jcumming@andrew.cmu.edu
Clifford Smyth
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139, USA. E-mail: csmyth@math.mit.edu
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Extract

Classical Ramsey theory (at least in its simplest form) is concerned with problems of the following kind: given a set X and a colouring of the set [X]n of unordered n-tuples from X, find a subset Y ⊆ X such that all elements of [Y]n get the same colour. Subsets with this property are called monochromatic or homogeneous, and a typical positive result in Ramsey theory has the form that when X is large enough and the number of colours is small enough we can expect to find reasonably large monochromatic sets.

Polychromatic Ramsey theory is concerned with a “dual” problem, in which we are given a colouring of [X]n and are looking for subsets Y ⊆ X such that any two distinct elements of [Y]n get different colours. Subsets with this property are called polychromatic or rainbow. Naturally if we are looking for rainbow subsets then our task becomes easier when there are many colours. In particular given an integer k we say that a colouring is k-bounded when each colour is used for at most k many n-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 write to mean “every colouring of [κ]n in k colours has a monochromatic set of order type α”. We will write to mean “every k-bounded colouring of [κ]n has a polychromatic set of order type α”. We note that when κ is infinite and k is finite a k-bounded colouring will use exactly κ colours, so we may as well assume that κ is the set of colours used.

Keywords

Polychromatic Ramsey theoryforcing axioms
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 865 - 896
Copyright
Copyright © Association for Symbolic Logic 2007

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