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Erd\\H{o}s and Gy\\'arf\\'as showed that $f(n, p, q)$ grows linearly in $n$ when $p$ is fixed and $q=q_{\\text{lin}}(p):=\\binom p2-p+3$. Similarly they showed that $f(n, p, q)$ is quadratic in $n$ when $p$ is fixed and $q=q_{\\text{quad}}(p):=\\binom p2-\\frac p2+2$. In this note we improve on the known estimates for $f(n, p, q_{\\text{lin}})$ and $f(n, p, q_{\\text{quad}})$. 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