Hypergraphs with arbitrarily small codegree Tur\'an density
Sim\'on Piga, Bjarne Sch\"ulke

TL;DR
This paper demonstrates that for any small positive number, there exists a hypergraph with arbitrarily small positive codegree Turán density, contrasting with classical Turán density limitations.
Contribution
It constructs hypergraphs with arbitrarily small positive codegree Turán density, showing the density can be made arbitrarily close to zero.
Findings
Existence of hypergraphs with arbitrarily small positive codegree Turán density.
Contrast with classical Turán density limitations.
Provides new insights into hypergraph extremal problems.
Abstract
Let . Given a -uniform hypergraph , the minimum codegree is the largest such that every -set of is contained in at least edges. Given a -uniform hypergraph , the codegree Tur\'an density of is the smallest such that every -uniform hypergraph on vertices with contains a copy of . Similarly as other variants of the hypergraph Tur\'an problem, determining the codegree Tur\'an density of a hypergraph is in general notoriously difficult and only few results are known. In this work, we show that for every , there is a -uniform hypergraph with . This is in contrast to the classical Tur\'an density, which cannot take any value in the interval due to a fundamental result by Erd\H{o}s.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Graph theory and applications
