Design Theory and Some Non-simple Forbidden Configurations

Abstract

Let 1_k 0_l denote the (k+l)\times 1 column of k 1's above l 0's. Let q. (1_k 0_l) $ denote the (k+l)xq matrix with q copies of the column 1_k0_l. A 2-design S_{\lambda}(2,3,v) can be defined as a vx(\lambda/3)\binom{v}{2} (0,1)-matrix with all column sums equal 3 and with no submatrix (\lambda+1).(1_20_0). Consider an mxn matrix A with all column sums in {3,4,... ,m-1}. Assume m is sufficiently large (with respect to \lambda) and assume that A has no submatrix which is a row permutation of (\lambda+1). (1_2 0_1). Then we show the number of columns in A is at most (\lambda)/3)\binom{m}{3} with equality for A being the columns of column sum 3 corresponding to the triples of a 2-design S_{\lambda}(2,3,m). A similar results holds for(\lambda+1). (1_2 0_2). Define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. Given two matrices A, F, we define A to have F as a configuration if and only if some submatrix of A is a row and column permutation of F. Given m, let forb(m,q.(1_k 0_l)) denote the maximum number of possible columns in a simple m-rowed matrix which has no configuration q.(1_k 0_l). For m sufficiently large with respect to q, we compute exact values for forb(m,q.(1_1 0_1)), forb(m,q.(1_2 0_1)), forb(m,q.(1_2 0_2)). In the latter two cases, we use a construction of Dehon (1983) of simple triple systems S_{\lambda}(2,3,v) for \lambda>1. Moreover for l=1,2, simple mxforb(m,q.(1_2 0_l)) matrices with no configuration q.(1_2 0_l) must arise from simple 2-designs S_{\lambda}(2,3,m) of appropriate \lambda. The proofs derive a basic upper bound by a pigeonhole argument and then use careful counting and Turan's bound, for large m, to reduce the bound. For small m, the larger pigeonhole bounds are sometimes the exact bound. There are intermediate values of m for which we do not know the exact bound.