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Uniquely -saturated Hypergraphs
In this paper we generalize the concept of uniquely -saturated graphs to
hypergraphs. Let denote the complete -uniform hypergraph on
vertices. For integers such that , a -uniform
hypergraph with vertices is uniquely -saturated if does
not contain but adding to any -set that is not a hyperedge
of results in exactly one copy of . Among uniquely
-saturated hypergraphs, the interesting ones are the primitive ones
that do not have a dominating vertex---a vertex belonging to all possible
edges. Translating the concept to the complements of these
hypergraphs, we obtain a natural restriction of -critical hypergraphs: a
hypergraph is uniquely -critical if for every edge ,
and has a unique transversal of size .
We have two constructions for primitive uniquely -saturated
hypergraphs. One shows that for and where , there
exists such a hypergraph for every . This is in contrast to the case
and where only the Moore graphs of diameter two have this property. Our
other construction keeps fixed; in this case we show that for any fixed
there can only be finitely many examples. We give a range for
where these hypergraphs exist. For the range is completely determined:
. For larger values of the upper end of
our range reaches approximately half of its upper bound. The lower end depends
on the chromatic number of certain Johnson graphs