 

Neil Hindman,
MariaRomina Ivan, and
Imre Leader
Some new results on monochromatic sums and products in the rationals
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Published: 
March 5, 2023. 
Keywords: 
Ramsey Theory, monochromatic sums and products. 
Subject [2020]: 
05D10, 11B13. 


Abstract
Our aim in this paper is to show that, for any k, there is a finite colouring of the set of rationals whose denominators contain only the first k primes such that no infinite set has all of its finite sums and products monochromatic. We actually prove a `uniform' form of this: there is a finite colouring of the rationals with the property that no infinite set whose denominators contain only finitely many primes has all of its finite sums and products monochromatic. We also give various other results, including a new short proof of the old result that there is a finite colouring of the naturals such that no infinite set has all of its pairwise sums and products monochromatic.


Acknowledgements
N/A


Author information
Neil Hindman:
Department of Mathematics
Howard University
Washington, DC 20059, USA
nhindman@aol.com
MariaRomina Ivan:
Department of Pure Mathematics and Mathematical Statistics
Centre for Mathematical Sciences
Wilberforce Road, Cambridge, CB3 0WB, UK
mri25@dpmms.cam.ac.uk
Imre Leader:
Department of Pure Mathematics and Mathematical Statistics
Centre for Mathematical Sciences
Wilberforce Road, Cambridge, CB3 0WB, UK
i.leader@dpmms.cam.ac.uk

