Abstract
We study (0, 1)-matrices which contain no triangles (submatrices of order 3 with row and column sums 2) previously studied by Ryser. Let the row intersection of row i and row j of some matrix, when regarded as a vector, have a 1 in a given column if both row i and row j do and a zero otherwise. For matrices with no triangles, columns sums ⩾2, we find that the number of linearly independent row intersections is equal to the number of distinct columns. We then study the extremal (0, 1)-matrices with no triangles, column sums ⩾2, distinct columns, i.e., those of size mx( m 2). The number of columns of column sum l is m − l + 1 and they form a ( l − 1)-tree. The (( m 2)) columns have a unique SDR of pairs of rows with 1's. Also, these matrices have a fascinating inductive buildup. We finish with an algorithm for constructing these matrices.
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