A tight 
-tree
is an 
-uniform hypergraph 
whose vertex
set is the union of its hyperedges 
, which can be ordered so that, for every 
, there are a vertex 
and an index 
satisfying
 |
Thus each hyperedge after the first introduces one new vertex and shares its other 
vertices with an earlier
hyperedge. A tight tree of uniformity 
with 
hyperedges has 
vertices. When 
, a tight tree is precisely an ordinary tree.
Tight trees occur in extremal hypergraph theory. Kalai's conjecture gives the sharp
upper bound for the number of hyperedges
in an 
-vertex 
-uniform hypergraph
that contains no specified tight tree.
