A new result on the problem of Zarankiewicz

Abstract

Zarankiewicz ( Colloq. Math. 2 (1951) , 301) raised the following problem: Determine the least positive integer z ( m , n , j , k ) such that each 0–1-matrix with m rows and n columns containing z ( m , n , j , k ) ones has a submatrix with j rows and k columns consisting entirely of ones. This paper improves a result of Hylten-Cavallius ( Colloq. Math. 6 (1958) , 59–65) who proved: [ k 2 ] 1 2 ⩽ lim n→∞ inf z(n, n, 2, k)n − 3 2 ⩽ lim n→∞ sup z(n, n, 2, k)n − 3 2 ⩽ (k − 1) 1 2 . We prove that lim n→∞ z(n, n, 2, k)n − 3 2 exists and is equal to (k − 1) 1 2 . For the special case where k = 2 resp. k = 3 this result was proved earlier by Kövari, Sos and Turan ( Colloq. Math. 3 (1954) , 50–57) resp. Hylten-Cavallius.