  We study extensions of Tur\'an Theorem in edge-weighted settings. A
particular case of interest is when constraints on the weight of an edge come
from the order of the largest clique containing it. These problems are
motivated by Ramsey-Tur\'an type problems. Some of our proofs are based on the
method of graph Lagrangians, while the other proofs use flag algebras. Using
these results, we prove several new upper bounds on the Ramsey-Tur\'an density
of cliques. Other applications of our results are in a recent paper of Balogh,
Chen, McCourt and Murley.
