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

TL;DR

This paper introduces entropy-based proofs for Turán's theorem, explores connections between entropic measures and hypergraph properties, and determines Turán densities for new hypergraph families called tents.

Contribution

It provides a novel entropic proof of a density version of Turán's theorem and determines Turán densities for tents, extending previous hypergraph results.

Findings

Entropy-based formulation of Turán's theorem

Connection between entropic quantities and hypergraph spectral measures

Turán density determined for tents hypergraphs

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.