A Survey of Forbidden Configuration Results

Let F be a k×` (0,1)-matrix. We say a (0,1)-matrix A has F as a configuration if there is a submatrix of A which is a row and column permutation of F . In the language of sets, a configuration is a trace and in the language of hypergraphs a configuration is a subhypergraph. Let F be a given k × ` (0,1)-matrix. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. The matrix F need not be simple. We define forb(m,F ) as the maximum number of columns of any simple m-rowed matrix A which do not contain F as a configuration. Thus if A is an m×n simple matrix which has no submatrix which is a row and column permutation of F then n ≤ forb(m,F ). Or alternatively if A is an m × (forb(m,F ) + 1) simple matrix then A has a submatrix which is a row and column permutation of F . We call F a forbidden configuration. The fundamental result is due to Sauer, Perles and Shelah, Vapnik and Chervonenkis. For Kk denoting the k×2k submatrix of all (0,1)-columns on k rows, then forb(m,Kk) = ( m k−1 ) + ( m k−2 ) + · · · ( m 0 ) . We seek asymptotic results for forb(m,F ) for a fixed F and as m tends to infinity . A conjecture of Anstee and Sali predicts the asymptotically best constructions from which to derive the asymptotics of forb(m,F ). The conjecture has helped guide the research and has been verified for k × ` F with k = 1, 2, 3 and for simple F with k = 4 as well as other cases including ` = 1, 2. We also seek exact values for forb(m,F ).