A new result on the problem of Zarankiewicz

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: $\text{k}\text{2}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{?}\text{lim}{}_{\mathrm{n\to \infty }}\text{inf}\text{z(n, n, 2, k)n}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>2</mtext>}}\text{?}\text{lim}{}_{\mathrm{n\to \infty }}\text{sup}\text{z(n, n, 2, k)n}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>2</mtext>}}\text{? (k ? 1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. We prove that $\text{lim}{}_{\mathrm{n\to \infty }}\text{z(n, n, 2, k)n}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>2</mtext>}}$ exists and is equal to $\text{(k ? 1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. 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.