Extremal graphs for odd wheels

For a graph (Formula presented.), the Turán number of (Formula presented.), denoted by ex (Formula presented.), is the maximum number of edges of an (Formula presented.) -vertex (Formula presented.) -free graph. Let (Formula presented.) denote the maximum number of edges not contained in any monochromatic copy of (Formula presented.) in a 2-edge-coloring of (Formula presented.). A wheel (Formula presented.) is a graph formed by connecting a single vertex to all vertices of a cycle of length (Formula presented.). The Turán number of (Formula presented.) was determined by Simonovits in 1960s. In this paper, we determine ex (Formula presented.) when (Formula presented.) is sufficiently large. We also show that, for sufficient large (Formula presented.), (Formula presented.) which confirms a conjecture posed by Keevash and Sudakov for odd wheels.