Avoiding Patterns in Matrices Via a Small Number of Changes

Let A = {A1, . . . , Ar} be a partition of a set {1, . . . , m} × {1, . . . , n} into r nonempty subsets, and A = (aij) be an m × n matrix. We say that A has a pattern A provided that aij = ai′j′ if and only if (i, j), (i, j) ∈ At for some t ∈ {1, . . . , r}. In this note we study the following function f defined on the set of all m × n matrices M with s distinct entries: f(M ;A) is the smallest number of positions where the entries of M need to be changed such that the resulting matrix does not have any submatrix with pattern A. We give an asymptotically tight value for f(m, n; s,A) = max{f(M ;A) : M is an m × n matrix with at most s distinct entries}.