Turán’s theorem inverted

In this note we complete an investigation started by Erdős in 1963 that aims to find the strongest possible conclusion from the hypothesis of Turán’s theorem in extremal graph theory. r + ( s 1 , … , s r ) be the complete r -partite graph with parts of sizes s 1 ≥ 2 , s 2 , … , s r with an edge added to the first part. Letting t r ( n ) be the number of edges of the r -partite Turán graph of order n , we prove that: l r ≥ 2 and all sufficiently small c > 0 , every graph of sufficiently large order n with t r ( n ) + 1 edges contains a K r + ( ⌊ c ln n ⌋ , … , ⌊ c ln n ⌋ , ⌈ n 1 − c ⌉ ) . o give a corresponding stability theorem and two supporting results of wider scope.