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

TL;DR

This paper extends Turán's theorem to weighted graphs with applications to Ramsey-Turán problems, using graph Lagrangians and flag algebras to derive new upper bounds on clique densities.

Contribution

It introduces weighted Turán theorems with novel bounds and methods, connecting them to Ramsey-Turán problems and expanding the theoretical framework.

Findings

New upper bounds on Ramsey-Turán density of cliques

Application of graph Lagrangians and flag algebras in proofs

Extensions of Turán's theorem to weighted settings

Abstract

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.