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We make small improvements on bounds of Erd\\H os and Gy\\'arf\\'as by showing ${5/6}n+1\\leq f(n,4,5)$ and for all even $n\\not\\equiv 1 \\pmod 3$, $f(n,4,5)\\leq n-1$ . For a complete bipartite graph $G=K_{n,n}$, we show an n-color construction to color the edges of $G$ so that every $C_4\\subseteq G$ is colored by at least three colors. This improves the best known upper bound of M. 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