A forbidden configuration theorem of Alon

Abstract

The paper studies the maximum possible number of distinct rows in a matrix with n columns with entries in column i in {0, 1, …, q i − 1} that does not contain certain forbidden submatrices. The results might have algorithmic significance if, for example, these matrices are the constraint matrix of a linear program. Combinatorial problems often yield a forbidden submatrix structure. Let (n; q 1, q 2, …, q n) -matrices be matrices on n columns with entries in column i in {0, 1, …, q i − 1} and let S be a family of subsets of {1, 2, …, n}. Let f( n, S) be the number of (n; q 1, q 2, …, q n) -rows which for each S ϵ S do not have 0's in all columns S. Noga Alon proved that if A is an m × n (n; q 1, q 2, …, q n) -matrix with no repeated rows, and for each S ϵ S, not all possible rows on columns S, then m ⩽ f( n, S). This paper provides an inductive proof and new f( n, S) × n matrices A as above. A linear algebra proof is given for the case q 1 = q 2 = … = q n = 2. Alon's shift proof technique is extended to handle the case A does not have all possible rows on S, each row occurring at least t times. Some other results concerning the extremal f( n, S) × n matrices are presented.