What is ex(n, H) (Turan Number)? I am struggling with understanding what exactly does $ex(n, K_{n,n})$ represent? It's defined as the turan number, but what exactly does this mean visually in terms of graphs? The formal definition that I have is $ex(n, K_{n,m}) = \max\{|E(G)|\cdot |V(G)| = n, K_{n,m} \text{ is not an induced subgraph of G}\}$. I understand the definition, but if I were given values of let's say $n = 50$ and $K_{n,m} = K_{3,5}$, how would I get to the $ex(50, K_{3,5})$?
Also, how is the Turan number related to the Turan graph? Would also be grateful if anyone could direct me to any resources where I can read more about that- books or videos?
Thank you!
 A: It is not actually easy to find any particular value $\mathrm{ex}(n,H)$. In your example, if we're looking for $\mathrm{ex}(50,K_{3,5})$, we are looking for a $50$-vertex graph with no $K_{3,5}$ subgraphs in it, and we want it to have as many edges as possible.
We can prove some lower bounds on this number without too much effort. For example, the graph $K_{2,48}$ has no copies of $K_{3,5}$ in it; since $K_{2,48}$ has $96$ edges, we know $\mathrm{ex}(50,K_{3,5}) \ge 96$. Of course, proving better lower bounds requires creativity.
Proving any kind of upper bound is tricky. Here's one approach that does not use sophisticated tools. Suppose we have a $50$-vertex graph $G$ with three vertices $u,v,w$ of degree at least $35$. There are at most $14$ vertices in the rest of the graph not adjacent to $u$, at most $14$ not adjacent to $v$, and at most $14$ not adjacent to $w$; delete them all, and we are still left with $u,v,w$ and $5$ more vertices, which must form a $K_{3,5}$. So if $G$ does not have a $K_{3,5}$, it can have at most $2$ vertices with degree $35$ or more. This gives us a degree sum of at most $2 \cdot 49 + 48 \cdot 34 = 1730$, for at most $865$ edges; we have shown that $\mathrm{ex}(50,K_{3,5}) \le 865$.
Of course, if all we can say is that $96 \le \mathrm{ex}(50,K_{3,5}) \le 865$, that's not saying much. Lots of work has been done to get better bounds (in general, not specifically for this example). For example, the Kővári–Sós–Turán theorem gives a bound of $O(n^{2-1/\min\{s,t\}})$ on $\mathrm{ex}(n,K_{s,t})$; I am not sure what the constant is, so I'm not sure what their bound gives when $n=50$.

The relationship between the Turán numbers and the Turán graph is that the Turán graph $T(n,r)$ turns out to be the $n$-vertex graph with the most edges that does not contain a $K_{r+1}$. So if $e(T(n,r))$ is the number of edges in the Turán graph, then $\mathrm{ex}(n,K_{r+1}) = e(T(n,r))$. This is approximately $(1 - \frac1r)\binom n2$.
