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An increasing path of length $k$ in $G$, denoted $I_k$, is a sequence of $k+1$ vertices $1 \\leq i_1 < i_2 < \\dots < i_{k+1}$ such that $i_1, i_2, \\ldots, i_{k+1}$ is a path in $G$. For $k \\geq 2$, let $p(k)$ be the supremum of $\\liminf_{ n \\rightarrow \\infty} \\frac{ e(G_n) }{n^2}$ over all $I_k$-free graphs $G$. In 1962, Czipszer, Erd\\H{o}s, and Hajnal proved that $p(k) = \\frac{1}{4} (1 - \\frac{1}{k})$ for $k \\in \\{2,3 \\}$. 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An increasing path of length $k$ in $G$, denoted $I_k$, is a sequence of $k+1$ vertices $1 \\leq i_1 < i_2 < \\dots < i_{k+1}$ such that $i_1, i_2, \\ldots, i_{k+1}$ is a path in $G$. For $k \\geq 2$, let $p(k)$ be the supremum of $\\liminf_{ n \\rightarrow \\infty} \\frac{ e(G_n) }{n^2}$ over all $I_k$-free graphs $G$. In 1962, Czipszer, Erd\\H{o}s, and Hajnal proved that $p(k) = \\frac{1}{4} (1 - \\frac{1}{k})$ for $k \\in \\{2,3 \\}$. 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