On a refinement of the Ahlswede--Katona Theorem

TL;DR

This paper refines the Ahlswede--Katona theorem by considering graphs with large independent sets where each vertex has high degree, motivated by hypergraph Turán problems.

Contribution

It provides a more precise bound on the maximum density of 2-edge stars under specific structural assumptions.

Findings

Refined bound for 2-edge star density in graphs with large independent sets

Application of the refinement to hypergraph Turán problems

Enhanced understanding of extremal graph configurations

Abstract

A classical theorem of Ahlswede and Katona determines the maximum density of the $2$-edge star in a graph with a given edge density. Motivated by its application in hypergraph Tur\'{a}n problems, we establish a refinement of their result under the additional assumption that the graph contains a large independent set in which every vertex has high degree.