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Exact Results on the Number of Restricted Edge Colorings for Some Families of Linear Hypergraphs


  • Contract grant sponsor: GIF; Contract grant number: I-889-182.6/2005 (to Y. P.).


For k-uniform hypergraphs F and H and an integer inline image, let inline image denote the number of r-colorings of the set of hyperedges of H with no monochromatic copy of F and let inline image, where the maximum is taken over the family inline image of all k-uniform hypergraphs on n vertices. Moreover, let inline image be the usual extremal function, i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F. Here, we consider the question for determining inline image for F being the k-uniform expanded, complete graph inline image or the k-uniform Fan(k)-hypergraph inline image with core of size inline image, where inline image, and we show

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for inline image and n large enough. Moreover, for inline image or inline image, for k-uniform hypergraphs H on n vertices, the equality inline image only holds if H is isomorphic to the ℓ-partite, k-uniform Turán hypergraph on n vertices, once n is large enough. On the other hand, we show that inline image is exponentially larger than inline image, if inline image.

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