http://ift.tt/29nbnjj A few weeks ago, I e-mailed Will Sawin excitedly to tell him that I could extend the new bounds on three-term arithmetic progression free subsets of to . Will politely told me that I was the third team to get there — he and Eric Naslund already had the result, as did Fedor Petrov. But I think there might be some expository benefit in writing up the three arguments, to see how they are all really the same trick underneath. Here is the result we are proving: Let be a prime power and let be the cyclic group of order . Let be a set which does not contain any three term arithmetic progression, except for the trivial progressions . Then The exciting thing about this bound is that it is exponentially better than the obvious bound of . Until recently, all people could prove was bounds like , and this is still the case if is not a prime power. All of our bounds extend to the colored version: Let be a list of triples in such that , but if are not all equal. ...