Balanced matrices and property (G)

Let A be a (0,1)-matrix with n rows and m columns, considered as the incidence matrix of a hypergraph H with edges E 1, E 2, …, E n (the columns) and with vertices x 1, x 2, …, x n (the rows). H is called a balanced hypergraph if A does not contain a square sub-matrix of odd order with exactly two ones in each row and in each column. In this paper, we prove more “minimax” equalities for balanced hypergraphs, than those already proved in Berge [1], Berge and Las Vergnas [3], Fulkerson et al. [7], Lovasz [12]; in fact, the known results will follow easily from our main theorem.