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

Weighted Tur\'an theorems with applications to Ramsey-Tur\'an type of problems