Definition
Let Ka,b denote a complete bipartite graph with a vertices on one side of the bipartition and b vertices on the other side. Define Za,b(m,n) to be the smallest integer k such that every bipartite graph that has m vertices on one side of its bipartition, n vertices on the other side, and k edges contains a subgraph isomorphic to Ka,b.
An alternative and equivalent definition is that Za,b(m,n) is the smallest integer k such that every (0,1)-matrix of size m × n with k 1's must have a set of a rows and b columns such that the corresponding a×b submatrix is made up only of 1's.
For the specific case when m = n and a = b the shorter notation Za(n) = Za,b(m,n) may also be used.
Read more about this topic: Zarankiewicz Problem
Famous quotes containing the word definition:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)