.. < /b> . eds ., < /b> SpringerVerlag, Heidelberg, (1996 ). < /b> [3] F.R.K.ChungandJ.L.Goldwasser,Maximumsubsetsof( 0, < /b> 1] with < /b> no < /b> solutions < /b> to< /b> x + < /b> y = kz, Electron. J. Combin. 3 (1996 ), < /b> R 1. < /b> [4] K. Roth, On certain .. < /b> . k-sum-free. b) |A< /b> i|≥ |A< /b> i−1 |. < /b> Proof. a)< /b> Clearly, it is enough to < /b> prove the claim for i ≤ t, so we may assume that s ≤ ri . < /b> Suppose there are a,< /b> b, c ∈ A< /b> i with < /b> a < /b> + < /b> b = kc. A< /b> iis of < /b> the form A< /b> i = A< /b> i−1∩ .. < /b> . findthat |B| = |A < /b> ∩ [1,w]|≤ |B ∩ [1,r2 ,B ]| + < /b> (k − 2)wk . < /b> (4)But s B = s A< /b> implies that r2 ,B ≤ r2 ,A < /b> , < /b> hence that B ∩ [1,r2 ,B ] ⊆ A < /b> ∩ [1,r2 ,A< /b> ]. < /b> Thus (3)and (4) yield the inequality |A|< /b> ≤|A...