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In 1980, Erd\\H{o}s and S\\'{o}s posed some questions about the growth of $\\Delta_s$. The best known concrete bounds on $\\Delta_s$ are $3 \\le \\Delta_s \\le s$, and they have not improved since the stating of the problem. In this paper we present some constructions, which imply in particular that $R(K_3,K_s) \\ge R(K_3,K_{s-1}-e) + 4$. 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