Paths of Odd Order in Graphs with Given Edge Density

We determine the asymptotic maximum number of unlabelled copies of $P_{2r+1}$ in graphs with prescribed edge density, where $r\ge1$ is fixed and $P_{2r+1}$ denotes the path on $2r+1$ vertices.

If an $n$ vertex graph $G$ has edge density $c=2e(G)/n^2$, then the maximum is $\frac12S_r(c)n^{2r+1}+O(n^{2r})$ for $0<c\le c_r$, and $\frac12c^{r+1/2}n^{2r+1}+O(n^{2r})$ for $c_r\le c<1$, where $S_r(c)$ is the value given by the quasi-star construction and $c_r\in(0,1)$ is an explicit algebraic transition point.

Thus the quasi-star construction is asymptotically extremal below the transition, while the quasi-clique construction is asymptotically extremal above the transition.

This extends the quasi-star versus quasi-clique theorem of Ahlswede and Katona for $P_3$ and the theorem of Nagy for $P_5$ to all paths with an odd number of vertices.

The proof reduces the problem to threshold graphons and then to two endpoint families.

The three-step endpoint is handled by reducing the required inequality to coefficient nonnegativity in a Bernstein expansion, which is proved by a direct combinatorial argument.