# A version of the Tur{\'a}n problem for linear hypergraphs-density conditions

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Halina Bielak

Inst. of Math., UMCS, Lublin

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**Minisymposium:**
VERTEX COLOURINGS AND FORBIDDEN SUBGRAPHS

**Content:**
Let $\mathcal{H}=(V,\mathcal{E})$ be a simple linear
hypergraph. We consider a blow-up hypergraph
$\mathcal{G}[\mathcal{H}]$ of $\mathcal{H}$. We are interested in
the following problem. We have to decide whether there exists a
blow-up hypergraph $\mathcal{B}[\mathcal{H}]$ of $\mathcal{H}$,
with given hyperedge densities, such that $\mathcal{H}$ is not a
transversal in $\mathcal{B}[\mathcal{H}]$. Recently the density
Tur{\'a}n problem was studied by Csikv{\'a}ri and Nagy for simple
graphs [{\em Combin. Prob. Comput.} 21 (2012) 531--553] and [{\em
The Electronic J. Combin.} 18 (2011) \# P46].
We generalize their results. We present an efficient algorithm to
decide whether a given set of hyperedge densities ensures the
existence of a linear hypertree $\mathcal{T}$ in all blow-up
hypergraphs $\mathcal{B}[\mathcal{T}]$. Moreover, we show some
conditions for critical hyperedge density of linear hypertrees and
some other linear hypergraphs.