**Journal of Graph Theory**

# 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.).

## Abstract

For *k*-uniform hypergraphs *F* and *H* and an integer , let denote the number of *r*-colorings of the set of hyperedges of *H* with no monochromatic copy of *F* and let , where the maximum is taken over the family of all *k*-uniform hypergraphs on *n* vertices. Moreover, let 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 for *F* being the *k*-uniform expanded, complete graph or the *k*-uniform Fan(*k*)-hypergraph with core of size , where , and we show

for and *n* large enough. Moreover, for or , for *k*-uniform hypergraphs *H* on *n* vertices, the equality 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 is exponentially larger than , if .