During World War II, Zarankiewicz took part in illegal teaching, forbidden by the German authorities, and eventually was sent to a concentration camp. He successfully survived and became a teacher at Warsaw University of Technology.
The Zarankiewicz problem is named after Zarankiewicz. This problem asks, for a given size of (0,1)-matrix, how many matrix entries must be set equal to 1 in order to guarantee that the matrix contains at least one a × b submatrix is made up only of 1's. An equivalent formulation in extremal graph theory asks for the maximum number of edges in a bipartite graph with no complete bipartite subgraph Ka,b.
The Zarankiewicz crossing number conjecture in the mathematical field of graph theory is also named after Zarankiewicz. The conjecture states that the crossing number of a complete bipartite graph equals
Zarankiewicz proved that this formula is an upper bound for the actual crossing number. The problem of determining the number was suggested by Paul Turán and became known as Turán's brick factory problem.