Specific cases of forbidden subgraph problem From Wikipedia - Forbidden subgraph problem:
Erdős–Stone theorem states that for all positive integers $n$ and all graphs $G$:
$$\operatorname{ex}(n,G)=\left(1-\frac{1}{\chi(G)-1}+o(1)\right)\frac{n^2}{2}$$
where $\operatorname{ex}(n,G)$ is the maximal number of edges that an $n$-vertex graph can have such that it does not have a subgraph isomorphic to $G$, and $\chi(G)$ is the chromatic number of $G$.
I would like to know about specific examples of graphs $G$ with $\chi(G) = 4$ and/or $\chi(G) = 3$, and about $n/2$ vertices or more, and as many edges as possible, such that we can say exactly (at least for values of $n$ greater than a certain value) that:
$$\operatorname{ex}(n,G) \le \frac{n^2}{3}$$
In other words, we know that any graph with $n$ vertices and at least $n^2/3$ edges must contain $G$.
 A: Well, if $\chi(G) = 3$, Erdős–Stone gives us $(1+o(1)) \frac{n^2}{4}$, which is guaranteed to be less than $\frac{n^2}{3}$ for $n$ above a certain value.
If $\chi(G) = 4$, and $n$ is a multiple of $3$, then $K_{n/3, n/3, n/3}$ is a graph with exactly $\frac{n^2}{3}$ edges which does not contain a copy of $G$. Therefore the only chance for $\text{ex}(n,G) \le \frac{n^2}{3}$ to hold is if adding a single edge to $K_{n/3,n/3,n/3}$ is guaranteed to create a copy of $G$; equivalently, if $G$ contains an edge $e$ such that $\chi(G-e) = 3$.
This is a necessary condition, but it is not clear to me that it is sufficient. However, Turán's theorem tells us that $\text{ex}(n,K_4)$ is the number of edges in nearly-balanced tripartite graph with $n$ vertices: $\text{ex}(n,K_4) = \frac{n^2}{3} - \frac{r(3-r)}{6}$, where $r = n \bmod 3$. In particular, $\text{ex}(n,K_4) \le \frac{n^2}{3}$ for all $n$, so $K_4$ is an example of a $4$-chromatic graph with this property.
Other somewhat derivative examples exist. For example, let $G$ be the $5$-vertex, $7$-edge graph obtained from $K_4$ by adding a leaf vertex. If an $n$-vertex graph contains $K_4$ but not $G$, then every copy of $K_4$ must be a connected component; if there are $k$ such copies, then the graph has at most $6k + \text{ex}(n-4k, K_4)$ edges, which is less than $\text{ex}(n,K_4)$ for large $n$.  Therefore, for large $n$, $\text{ex}(n,G) = \text{ex}(n,K_4) \le \frac{n^2}{3}$.
It would be interesting to see if $\text{ex}(n,G) \le \frac{n^2}{3}$ when, for example, $G$ is a wheel graph with an even number of vertices. Then $\chi(G) = 4$, and $G$ satisfies the necessary condition given earlier, so it is possible that $\text{ex}(n,G) \le \frac{n^2}{3}$. But we cannot get there by piggybacking off of Turán's theorem, because $G$ does not contain a $K_4$.
