Turán's Theorem is a fundamental result in graph theory that provides a way to determine the maximum number of edges in a graph that does not contain a complete subgraph of a given size. Specifically, it states that for any integer r , the maximum number of edges in a graph with n vertices that does not contain a complete subgraph K_r+1 is given by a specific formula involving n and r . This theorem is particularly useful in combinatorial mathematics and has applications in various fields, including computer science and network theory. It helps researchers understand the structure of graphs and the relationships between their vertices, guiding the design of networks that avoid certain configurations.