When entropy meets Turán: new proofs and hypergraph Turán results
Ting-Wei Chao, Hung-Hsun Hans Yu
TL;DR
The paper addresses hypergraph Turán densities by introducing an entropic reformulation of the density Turán theorem and developing a counting framework that scalably handles stars of all sizes. By relating entropy to blowup density and, in a generalized sense, to the spectral radius, the authors prove a new entropic Turán theorem and leverage it to determine the Turán density for a new family of hypergraphs called tents, including Delta_\lambda with |\lambda|=k and ell(\lambda)=2, where π(\mathcal{F}_k) = k!/k^k. Two proofs of the main entropic-tent result are given, one via a direct entropy-inequality argument and another via a partial-forest/mixing approach, illustrating the method's robustness and potential for broader hypergraph Turán results. The work also ties entropy to Lagrangian and spectral radius, discusses partial-hypergraph frameworks, and outlines future directions like entropic flag algebras and stability analyses, broadening the toolkit for extremal hypergraph problems.
Abstract
In this paper, we provide a new proof of a density version of Turán'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 Turán 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.