More on a Problem of Zarankiewicz

Details More on a Problem of Zarankiewicz More on a Problem of Zarankiewicz

2012-01-06

arxiv.org/abs/1201.1377v1

Computer Science

Discrete Mathematics

14 pages

Scientific paper

We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical results in graph theory: the theorem of K\H{o}v\'{a}ri, S\'{o}s and Tur\'{a}n on the minimum number of edges in a bipartite graph that has no large independent set, and the theorem of Hansel (also Katona and Szemer\'{e}di and Krichevskii) on the sum of the sizes of bipartite graphs that can be used to construct a graph (non-necessarily bipartite) that has no large independent set. Our results unify the underlying combinatorial principles developed in the proof of tight lower bounds for depth-two superconcentrators.

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