A Note on the Number of Triangles in Graphs Without the Suspension of a Path on Four Vertices

Abstract

The suspension of the path P 4 consists of a P 4 and an additional vertex connected to each of the four vertices, and is denoted by ˆ P 4 . The largest number of triangles in a ˆ P 4 -free n -vertex graph is denoted by ex( n, K 3 , ˆ P 4 ). Mubayi and Mukherjee in 2020 showed that ex( n, K 3 , ˆ P 4 ) = n 2 / 8 + O ( n ). We show that for sufficiently large n , ex( n, K 3 , ˆ P 4 ) = ⌊ n 2 / 8 ⌋ .