A path Turan problem for infinite graphs

Let $G$ be an infinite graph whose vertex set is the set of positive integers, and let $G_n$ be the subgraph of $G$ induced by the vertices ${1,2, \dots , n }$\mathrm{.}Anincreasingpathoflength$k$in$G$,denoted$I_k$, is a sequence of $k+1$ vertices $1\le i_1<i_2<\cdots <i_{k+1}$ such that $i_1, i_2, \ldots, i_{k+1}$isapathin$G$\mathrm{.}For$k \geq 2$,let$p(k)$ be the supremum of ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}\frac{e(G_n)}{n^2}$ 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 $k\in \{2,3\}$. Erd\H{o}s conjectured that this holds for all $k\ge 4$. This was disproved for certain values of $k$ by Dudek and R"{o}dl who showed that $p(16) > \frac{1}{4} (1 - \frac{1}{16})$and$p(k) > \frac{1}{4} + \frac{1}{200}$forall$k \geq 162$. Given that the conjecture of Erd\H{o}s is true for $k\in \{2,3\}$ but false for large $k$, it is natural to ask for the smallest value of $k$ for which $p(k)>\frac{1}{4}(1-\frac{1}{k})$. In particular, the question of whether or not $p(4)=\frac{1}{4}(1-\frac{1}{4})$ was mentioned by Dudek and R"{o}dl as an open problem. We solve this problem by proving that $p(4) \geq \frac{1}{4} (1 - \frac{1}{4} ) + \frac{1}{584064}$and$p(k) > \frac{1}{4} (1 - \frac{1}{k})$for$4 \leq k \leq 15$\mathrm{.}Wealsoshowthat$p(4) \leq \frac{1}{4}$whichimprovesuponthepreviouslybestknownupperboundon$p(4)$\mathrm{.}Therefore,$p(4)$mustliesomewherebetween$\frac{3}{16} + \frac{1}{584064}$and$\frac{1}{4}$