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