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Thursday 7 April 2022

# Bounding monochromatic arithmetic progressions

Speaker: Dr Jonathan Kariv,
Time: 13:00
Venue: Online only

We find a $$2$$-colouring of the integers $${1,2,3,,…,n}$$ that minimizes the number of monochromatic arithmetic $$3$$-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers that are all the same colour. Previous work by Parillo, Robertson and Saracino conjectured an optimal colouring for large $$n$$ that involves $$12$$ coloured blocks. Here, we prove that the conjecture is optimal among anti-symmetric colorings with $$12$$ or fewer coloured blocks. We leverage a connection to the colouring the continuous interval $$[0,1]$$ first identified by Butler, Costello and Graham. Our proof relies on identifying classes of colorings with permutations using mixed integer linear programming. Joint work with Torin Greenwood and Noah Williams.