Threshold functions for the bipartite Turán property

Document Type

Article

Publication Date

1-1-1997

Abstract

Let G2(n) denote a bipartite graph with n vertices in each color class, and let z(n, t) be the bipartite Turán number, representing the maximum possible number of edges in G2(n) if it does not contain a copy of the complete bipartite subgraph K(t, t). It is then clear that ζ(n, t) - n2 - z(n, t) denotes the minimum number of zeros in an n × n zero-one matrix that does not contain a t × t submatrix consisting of all ones. We are interested in the behaviour of z(n, t) when both t and n go to infinity. The case 2 ≤ t ≪ C n1/5 has been treated in [9]; here we use a different method to consider the overlapping case log n ≪C t ≪ C n1/3. Fill an n × n matrix randomly with z ones and ζ = n2 - z zeros. Then, we prove that the asymptotic probability that there are no t × t submatrices with all ones is zero or one, according as z ≥ (t/ne)2/t exp{an/t2} or z ≤ (t/ne)2/t exp{(logt - bn)/t2}, where an tends to infinity at a specified rate, and bn → ∞ is arbitrary. The proof employs the extended Janson exponential inequalities [1].

Publication Title

Electronic Journal of Combinatorics

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