When entropy meets Tur\'an: new proofs and hypergraph Tur\'an results

Abstract

In this paper, we provide a new proof of a density version of Tur\'an's theorem. We also rephrase both the theorem and the proof using entropy. With the entropic formulation, we show that some naturally defined entropic quantity is closely connected to other common quantities such as Lagrangian and spectral radius. In addition, we also determine the Tur\'an density for a new family of hypergraphs, which we call tents. Our result can be seen as a new generalization of Mubayi's result on the extended cliques.