On a refinement of the Ahlswede--Katona Theorem

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 Turán 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.

💡 Research Summary

The paper revisits the classical Ahlswede–Katona theorem, which determines the maximum possible number of 2‑edge stars (also called cherries) in a graph with a prescribed edge density. While the original result shows that the extremal graphs are either a quasi‑star (the complement of a dense clique) or a quasi‑clique, those constructions do not contain a large independent set in which every vertex has high degree—a property that is essential in several hypergraph Turán problems.

To address this gap the authors introduce a new family of graphs G(n,m,ℓ,k). Here n is the number of vertices, m the number of edges, ℓ the size of an independent set I, and k a lower bound on the degree of every vertex in I (i.e., δ_G(I) ≥ k). The central problem (Problem 1.2) asks for the maximum number of cherries N(S₂,G) among all graphs in G(n,m,ℓ,k).

The paper first provides lower bounds by explicit constructions: the quasi‑star S(n,m) (which already satisfies the independent‑set condition when m is large enough), and two hybrid graphs G₁(n,m,ℓ,k) and G₂(n,m,ℓ,k) that combine a dense core with a bipartite “high‑degree” part attached to the independent set. These yield the lower‑bound Fact 1.3 for the asymptotic cherry density I(S₂,ρ,α,β) where ρ is the edge density, α≈ℓ/n and β≈k/n.

The main contributions are two exact asymptotic formulas. Theorem 1.4 treats the case α=β (the independent set size and the degree bound are equal) and shows that for edge densities ρ in the interval