4.33 Given that in a red-blue colored $K_{8}$ there is neither a red triangle nor a blue $K_{4}$, how many red edges are there at minimum? What is the maximum number of red edges? Explain your reasoning.
[Solution] In $K_{8}$, each point is connected by 7 edges. If vertex $A$ leads to 6 blue edges, then in the $K_{6}$ formed by the other endpoints of these 6 edges, there is either a red triangle or a blue triangle, which would lead to a blue $K_{4}$, contradicting the given condition. Therefore, each vertex in $K_{8}$ must lead to at least two red edges, resulting in at least 8 red edges in $K_{8}$.

On the other hand, if vertex $B$ leads to 4 red edges, then in the $K_{4}$ formed by the other 4 vertices, there is either 1 red edge, leading to a red triangle, or it is a blue $K_{4}$, contradicting the given condition. Therefore, each vertex in $K_{8}$ can lead to at most 3 red edges, resulting in at most 12 red edges in $K_{8}$.
(1) If $K_{8}$ has exactly 8 red edges, then each point leads to exactly two red edges, so the 8 red edges must form one or two cycles. Since each point can lead to at most 3 red edges, if there are two cycles, the vertices do not intersect. Also, since there are no red triangles in the graph, if there are two cycles, each must have 4 edges.

When the 8 red edges form a cycle $A_{1} A_{2} A_{3} \cdots A_{8}$, $A_{1} A_{3} A_{5} A_{7}$ is a blue $K_{4}$; when the 8 red edges form two cycles $A_{1} A_{2} A_{3} A_{4}$ and $A_{5} A_{6} A_{7} A_{8}$, $A_{1} A_{3} A_{5} A_{7}$ is also a blue $K_{4}$. Therefore, $K_{8}$ must have at least 9 red edges.
(2) Suppose $K_{8}$ has exactly 9 red edges. Since there are no red triangles and each point can lead to at most 3 red edges and at least 2 red edges, among the 8 vertices, exactly 2 vertices each lead to 3 red edges, and the remaining 6 vertices each lead to 2 red edges. In this case, $K_{8}$ can only have the following two possible scenarios: the diagrams only show the red edges, where the dashed line in diagram (a) indicates that edge $A_{2} A_{7}$ can be removed and replaced with $A_{2} A_{6}$. It is easy to see that both diagrams (a) and (b) contain a blue $K_{4}: A_{1} A_{3} A_{5} A_{7}$. Therefore, $K_{8}$ must have at least 10 red edges.

In diagrams (c) and (d), there are neither red triangles nor blue $K_{4}$, and diagram (c) has 10 red edges, while diagram (d) has 12 red edges.
In summary, $K_{8}$ has at least 10 red edges and at most 12 red edges.