A Survey of Forbidden Configuration Results (Draft)

This paper surveys various results concerning the problem of forbidden configurations (also known as trace in the language of set systems). Let F be a k × ` (0,1)-matrix (the forbidden configuration). 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 . Perhaps the most fundamental result is due to Sauer, Perles and Shelah, Vapnik and Chervonenkis but there are many other results catalogued here. In the language of sets, a configuration is a trace. We seek exact values for forb(m,F ) as well as seeking 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 ). This conjecture has helped guide the research and has been verified for k×` F with k = 1, 2, 3, with ` = 1, 2 and for simple F with k = 4 as well as other cases.