A structural result for hypergraphs with many restricted edge colorings
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- 2010
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For k-uniform hypergraphs \cF and \cH and an integer r≥2,let cr,\cF(\cH) denote the number ofr-colorings of the set of hyperedges of \cH with no monochromaticcopy of \cF andlet cr,\cF(n)=max\cH∈\ccHncr,\cF(\cH),where the maximum runs over the family Hn of allk-uniform hypergraphs on n vertices. Moreover, let \ex(n,\cF) bethe usual \emph{Turán function}, i.e., the maximumnumber of hyperedges of an n-vertex k-uniform hypergraph whichcontains no copy of~\cF
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In this paper, we consider the question for determining cr,F(n)
for arbitrary fixed hypergraphs \cF and show%cr,\cF(n)=r\ex(n,\cF)+o(nk)%for r=2,3. Moreover, we obtain a structural result for r=2,3 and any\cH with cr,\cF(\cH)≥r\ex(|V(\cH)|,\cF) under the assumptionthat a stability result for the k-uniform hypergraph \cF exists and|V(H)| is sufficiently large. We also obtain exact resultsfor cr,\cF(n) when \cF is a 3- or 4-uniform generalized triangleand r=2,3, while cr,\cF(n)≫r\ex(n,\cF) for r≥4and n sufficiently large.
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For k-uniform hypergraphs \cF and \cH and an integer r≥2,let cr,\cF(\cH) denote the number ofr-colorings of the set of hyperedges of \cH with no monochromaticcopy of \cF andlet cr,\cF(n)=max\cH∈\ccHncr,\cF(\cH),where the maximum runs over the family Hn of allk-uniform hypergraphs on n vertices. Moreover, let \ex(n,\cF) bethe usual \emph{Turán function}, i.e., the maximumnumber of hyperedges of an n-vertex k-uniform hypergraph whichcontains no copy of~\cF
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- oai:www.edit.fis.uni-hamburg.de:publications/87f03a0a-e42e-4542-9301-b6f1b25065ef