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Let 1_k 0_l denote the (k+l)\times 1 column of k 1's above l 0's. Let q. (1_k0_l) $ denote the (k+l)xq matrix with q copies of the column 1_k0_l. A 2-designS_{\lambda}(2,3,v) can be defined as a vx(\lambda/3)\binom{v}{2} (0,1)-matrixwith 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 issufficiently large (with respect to \lambda) and assume that A has no submatrixwhich is a row permutation of (\lambda+1). (1_2 0_1). Then we show the numberof columns in A is at most (\lambda)/3)\binom{m}{3} with equality for A beingthe columns of column sum 3 corresponding to the triples of a 2-designS_{\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 repeatedcolumns. Given two matrices A, F, we define A to have F as a configuration ifand 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 asimple m-rowed matrix which has no configuration q.(1_k 0_l). For msufficiently large with respect to q, we compute exact values for forb(m,q.(1_10_1)), forb(m,q.(1_2 0_1)), forb(m,q.(1_2 0_2)). In the latter two cases, weuse 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 withno configuration q.(1_2 0_l) must arise from simple 2-designsS_{\lambda}(2,3,m) of appropriate \lambda.  The proofs derive a basic upper bound by a pigeonhole argument and then usecareful counting and Turan's bound, for large m, to reduce the bound. For smallm, the larger pigeonhole bounds are sometimes the exact bound. There areintermediate values of m for which we do not know the exact bound.

Design Theory and Some Non-simple Forbidden Configurations