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:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“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)