Colin Desmarais, 2017/03/31
Seminar series:  Combinatorics
Presenter:Colin Desmarais
Department:Mathematics
Institution:University of Manitoba
Date:Friday, March 31st, 2017
Time::
Location:111 Armes

Colin Desmarais, 2017/03/31

What is today called arithmetic Ramsey theory has its origins with Hilbert's cube lemma of 1892, Schur's theorem of 1916, and van der Waerden's theorem of 1927; all of which predate Ramsey's publication of his now famous theorem in 1930. These results guarantee a monochromatic solution to a particular system of equations in any finite colouring of the positive integers: for example, Schur's theorem states that every finite colouring of the positive integers has a monochromatic solution to x+y=z. Such an equation or system of equations is called partition regular.

In this talk I will give an overview of early results in arithmetic Ramsey theory, including Rado's 1933 characterization of partition regular systems. I will also discuss Deuber's proof in 1973 of a conjecture of Rado's that in any finite colouring of the positive integers some colour class contains solutions to every partition regular system. Finally, I will briefly state results in infinite and nonlinear partition regular systems.




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