A *triangle-free* graph is an undirected graph in which no three vertices form a triangle of edges. The *independence number* $α = α(G)$ of a graph $G$ is the cardinality of a maximum in dependent set of vertices.

The following theorem is well-known.

**Mantel’s theorem.** Let $G$ be a triangle-free graph with $n$ vertices. Then $G$ contains at most $\lfloor \frac{n^2}{4} \rfloor$ edges. Furthermore, the only triangle-free graph with $\lfloor \frac{n^2}{4} \rfloor$ edges is the complete graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil }$.

We observe that the unique extremal graph ($K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil }$) has independence number $α =\lceil \frac{n}{2} \rceil $. Therefore, I would like to inquire if the following question has been investigated.

**Problem.** Let $G$ be a triangle-free graph with $n$ vertices and $α(G) = k $ where $k\ne \lceil \frac{n}{2} \rceil $. Then what is the upper bound on the number of edges in $G$?

For small values of $k$, such as $k= 3, 4, 5$, is there an exact solution to this problem?

Has this problem been studied before?