Contributions to the problem of Zarankiewicz

Abstract

Let G 2 ( 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 G 2 ( n ) if it is not to contain a copy of the complete bipartite subgraph K ( t , t ). It is then clear that ζ ( n , t )= n 2 − z ( n , t ) denotes the smallest possible number of zeros in an n × n zero–one matrix that does not contain a t × t submatrix consisting of all ones. We investigate the behaviour of z ( n , t ) when n goes to infinity (and perhaps t →∞ as well). The case t= o (n 1/3 );t⪢ log n has been considered by Godbole et al. [Electron. J. Combin. 4 (1997) 14pp], and we focus here on the overlapping case 2⩽ t ⪡ n 1/5 . Fill an n × n matrix randomly with z ones and n 2 − z = ζ zeros. Then, we prove that if t →∞, the asymptotic probability that there are no t × t submatrices with all ones is one or zero, according as z=n 2−2/t t e 2/t {1+ o (1)} or z=n 2−2/t t e 2/t {1+ o ∗ (1)}, where the o(1) and o ∗ (1) functions may be taken to be ( log t−a n )/t 2 and ( log t+a n )/t 2 respectively, for any a n that tends to infinity. If t is finite, a less sharp result is shown to be valid. The proof employs the Janson exponential inequalities. Finally, the Stein–Chen method is employed to derive a Poisson approximation for the probability distribution of the number of copies of K ( t , t ) in the random bipartite graph; an extreme value limit for the probability that there are no t × t submatrices with all ones, when the value of z is at the threshold level, is derived as a corollary.