Dense neighbourhoods and Turán's theorem
We prove the following extension of Turán's theorem, conjectured by Erdös. Let tr(n) be the number of edges in the r-partite Turán graph Tr(n) of order n, and suppose G is a graph of order n with at least tr(n) edges. Then either G=Tr(n) or else there is a vertex x such that the subgraph spanned by the neighbours of x contains at least tr-1(d)+1 edges, where d is the degree of x. Furthermore d>crn, where cr is a constant. © 1981 Academic Press, Inc. All Rights Reserved.
Journal of Combinatorial Theory, Series B
Bollobás, B., & Thomason, A. (1981). Dense neighbourhoods and Turán's theorem. Journal of Combinatorial Theory, Series B, 31 (1), 111-114. https://doi.org/10.1016/S0095-8956(81)80016-0