Rainbow Arithmetic Progressions and Anti-Ramsey Results

The van der Waerden theorem in Ramsey theory states that, for every <formula id="ffm001"><formtex notation="AMSTeX">$k$</formtex></formula> and <formula id="ffm002"><formtex notation="AMSTeX">$t$</formtex></formula> and sufficiently large <formula id="ffm003"><formtex notation="AMSTeX">$N$</formtex></formula>, every <formula id="ffm004"><formtex notation="AMSTeX">$k$</formtex></formula>-colouring of <formula id="ffm005"><formtex notation="AMSTeX">$[N]$</formtex></formula> contains a monochromatic arithmetic progression of length <formula id="ffm006"><formtex notation="AMSTeX">$t$</formtex></formula>. Motivated by this result, Radoičić conjectured that every equinumerous 3-colouring of <formula id="ffm007"><formtex notation="AMSTeX">$[3n]$</formtex></formula> contains a 3-term rainbow arithmetic progression, <e1>i.e.</e1>, an arithmetic progression whose terms are coloured with distinct colours. In this paper, we prove that every 3-colouring of the set of natural numbers for which each colour class has density more than 1/6, contains a 3-term rainbow arithmetic progression. We also prove similar results for colourings of <formula id="ffm008"><formtex notation="AMSTeX">$\mathbb{Z}_n$</formtex></formula>. Finally, we give a general perspective on other <e1>anti-Ramsey-type</e1> problems that can be considered.