Bounding monochromatic arithmetic progressions
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.