Basic extremal graph theory
Extremal graph theory is devoted to, as one can guess, extremal questions concerning graphs. We start with some basic definitions.
Prerequisites: basic graph theory
Definitions recap: Recall that a graph $G$ is a pair $(V,E)$, where $V$ is a set and $E$ is a family of some pairs of elements from $V$. For finite graphs, one can say that $V = \{1,\ldots, n\}$, or $[n]$ for shorthand. The elements of $V$ are called vertices, and the elements of $E$ are called edges. Two graphs are isomorphic if we can obtain one from the other by simply relabeling the vertices. Formally, $G_1 = (V_1,E_1)$ and $G_2 = (V_2,E_2)$ are isomorphic if there is a bijection $\sigma: V_1\to V_2$ such that $(v_1,w_1)\in E_1$ if and only if $(\sigma(v_1),\sigma(w_1))\in E_2$.
We say that $H$ is a subgraph of $G$ if it is isomorphic to some graph $H'$ that can be obtained from $G$ by deleting both vertices and edges. Similarly, we say that $H$ is an induced subgraph of $G$ if it is isomorphic to a graph $H'$ that can be obtained from $G$ by deleting vertices only (i.e., there is a subset $V'$ of $V$ such that $H'=(V',E')$, and $E'$ consists of all pairs from $E$ that are contained in $V'$).
The first and key question of extremal graph theory is as follows: for a given graph $H$, determine the largest number of edges in an $n$-vertex graph that has no subgraph, isomorphic to $H$. This value is denoted $ex(n,H)$. We may also be interested to know, which graphs attain the extremal value.
The first examples of results of such kind are Mantel and Turan theorems that deal with the case when $H$ is a triangle and, more generally, a complete graph on $r$ vertices. The extremal example here is a complete multipartite graph with $r-1$ parts and which sizes are nearly equal, which means that they differ by at most one. (FIGURE?)
Here is an introductory video on Mantel's theorem.
Here is a video on Turan's theorem, that also gives an introduction to the field of extremal graph theory, from a course by Luke Postle:
Next, there is a comprehensive, but more demanding, video on both Mantel's and Turan's theorem that is a part of an excellent course by Yufei Zhao.
In the end of the video above, Yufei also mentions results for general graphs. Most importantly, the quantity $ex(n,H)$ is very well understood for graphs $H$ that have chromatic number $r\ge 3$. In this case, for a fixed $H$ and $n$ that tends to infinity, we have $$ex(n,H) = (1\frac 1{r-1}+o(1)){n\choose 2} \sim ex(n, K_r).$$
The case of bipartite graphs is much harder, and our understanding of $ex(n,H)$ is far from being complete in this case. There are very few examples in which we know the order of growth of this function. One such case when $H$ is a $4$-cycle. The upper bound is a particular case of the Kovari-Sos-Turan theorem gives a good bound for any complete bipartite graph $K_{s,t}$.
Here is a video from the course by Yufei Zhao.
Concluding, we know the asymptotics $ex(n,H)$ for graphs $H$ with $\chi(H)\ge 3$, and for such graphs this value is always quadratic. For all bipartite graphs $H$ we know that $ex(n,H) = O(n^{2-\epsilon})$ with some positive $\epsilon$ that depends on $H$. The exact value of this $\epsilon$, however, is known for very few graphs. One of the reasons for this is that it is difficult to find good constructions of $H$-free graphs in order to improve lower bounds.