A probabilistic threshold for monochromatic arithmetic progressions

Let fr(k)=k⋅rk/2 (where r≥2r≥2 is fixed) and consider r  -colorings of [1,nk]={1,2,…,nk}[1,nk]={1,2,…,nk}. We show that fr(k)fr(k) is a threshold function for k  -term arithmetic progressions in the following sense: if nk=ω(fr(k))nk=ω(fr(k)), then limk→∞⁡P([1,nk]limk→∞⁡P([1,nk] contains a monochromatic k-term arithmetic progression)=1k-term arithmetic progression)=1; while, if nk=o(fr(k))nk=o(fr(k)), then limk→∞⁡P([1,nk] contains ak-term monochromatic arithmetic progression)=0.