For any integer polynomial P with P(0)=0, any subset of [N] avoiding x, x+P(y), ..., x+kP(y) has size at most N (log log log N)^{-c}, with stronger bounds when P'(0)!=0.
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On polynomial progressions via transference
For any integer polynomial P with P(0)=0, any subset of [N] avoiding x, x+P(y), ..., x+kP(y) has size at most N (log log log N)^{-c}, with stronger bounds when P'(0)!=0.