Hypergraphs with arbitrarily small codegree Turán density

2023/07/06 by Simón Piga, Piga, Simón, Bjarne Schülke +1 · 1 citation
Mathematics · #Limits and Structures in Graph Theory #Graph theory and applications

paper · pdf · doi:10.48550/arxiv.2307.02876

Abstract

Let k≥ 3. Given a k-uniform hypergraph H, the minimum codegree δ(H) is the largest d∈ℕ such that every (k-1)-set of V(H) is contained in at least d edges. Given a k-uniform hypergraph F, the codegree Turán density γ(F) of F is the smallest γ∈ [0,1] such that every k-uniform hypergraph on n vertices with δ(H)≥ (γ+ o(1))n contains a copy of F. Similarly as other variants of the hypergraph Turán problem, determining the codegree Turán density of a hypergraph is in general notoriously difficult and only few results are known. In this work, we show that for every ε>0, there is a k-uniform hypergraph F with 0<γ(F)<ε. This is in contrast to the classical Turán density, which cannot take any value in the interval (0,k!/k^k) due to a fundamental result by Erdős.

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