Mantel's Theorem is a result in graph theory that states that in any triangle-free graph with n vertices, the maximum number of edges it can have is \fracn^24 . A triangle-free graph is one that does not contain any three vertices that are mutually connected by edges. This theorem helps in understanding the structure and limitations of certain types of graphs. The theorem was proven by Thomas Mantel in 1907 and is a special case of a more general result known as Turán's Theorem. It provides insight into how graphs can be constructed without forming triangles, which has implications in various fields, including computer science and combinatorics.