3. Given 10 points in space, where no four points lie on the same plane, connect some of the points with line segments. If the resulting figure contains no triangles and no spatial quadrilaterals, determine the maximum number of line segments that can be drawn.


Given a simple graph $G$ of order 10 with these 10 points as vertices and the connected line segments as edges. Let the vertices of graph $G$ be $v_{1}, v_{2}, \cdots, v_{10}$, and there are $k$ edges. Use $\operatorname{deg}\left(v_{i}\right)$ to denote the degree of vertex $v_{i}$.
If $\operatorname{deg}\left(v_{i}\right) \leqslant 3$ for $i=1,2, \cdots, 10$, then $k=\frac{1}{2} \sum_{i=1}^{10} \operatorname{deg}\left(v_{i}\right) \leqslant \frac{1}{2} \times 10 \times 3=15$; if there exists a vertex $v_{i}$ such that $\operatorname{deg}\left(v_{i}\right) \geqslant 4$, without loss of generality, let $\operatorname{deg}\left(v_{1}\right)=n \geqslant 4$, and $v_{1}$ is adjacent to $v_{2}, v_{3}, \cdots, v_{n+1}$. Thus, there are no edges between $v_{2}, v_{3}, \cdots, v_{n+1}$, otherwise, a triangle would be formed, which is a contradiction. Therefore, there are exactly $n$ edges between $v_{1}, v_{2}, \cdots, v_{n+1}$.
For each $v_{j}(n+2 \leqslant j \leqslant 10)$, $v_{j}$ can be adjacent to at most one of $v_{2}, v_{3}, \cdots, v_{n+1}$ (otherwise, if $v_{j}$ is adjacent to $v_{s}, v_{t}(2 \leqslant s<t \leqslant n+1)$, then $v_{1}, v_{s}, v_{j}, v_{t}$ would form a quadrilateral, which is a contradiction). Thus, there are at most $10-(n+1)=9-n$ edges between $v_{2}, v_{3}, \cdots, v_{n+1}$ and $v_{n+2}, \cdots, v_{10}$.
Among the $9-n$ vertices $v_{n+2}, \cdots, v_{10}$, since there are no triangles, by Turán's theorem (a simple $n$-order graph without triangles has at most $\left[\frac{n^{2}}{4}\right]$ edges), there are at most $\left[\frac{(9-n)^{2}}{4}\right]$ edges. Therefore, the number of edges in graph $G$ is
$$
k \leqslant n+9-n+\left[\frac{(9-n)^{2}}{4}\right] \leqslant 9+\left[\frac{25}{4}\right]=15 .
$$

The graph shown on the right has 15 edges and meets the requirements.
In conclusion, the maximum number of edges is 15.