The prompt is to find the number of edges of graph with $36$ vertices given that from every $4$ vertices, at least $2$ of them have to have an edge between them. We must prove that the graph G has at least 105 edges or find some non-trivial lower limit on the number of edges.
If we join $2$ vertices of all the $36$ vertices, we are left with $18$ edges, how is this possible, how should I go on about solving a problem like this? I know that $2|E| = deg(v)$ by handshake theorem, how should I find the degree of each vertex here?
Consider the set of all subgraphs of $G$ that have four vertices. There are ${36\choose 4}$ such graphs. For each such subgraph $H$, count the number of edges $e(H)$ in it. Then calculate $\sum_H e(H)$ in two ways. One way is to note that every edge appears ${34 \choose 2}$ many times. So $\sum_H e(H)=e(G) \times {34 \choose 2}$. On the other hand, each subgraph $H$ has at least one edge, so $\sum_H e(H) \geq {36 \choose 4}$. In other words,
$$e(G) \times {34 \choose 2} \geq {36 \choose 4}.$$
Simplifying both sides gives $e(G) \geq 105$.