In this paper we relate t-designs to a forbidden configuration problem in extremal set theory. Let 1t0l denote a column of t 1's on top of l 0's. Let q.1t0l denote the (t+l)xq matrix consisting of t rows of q 1's and l rows of q 0's. We consider extremal problems for matrices avoiding certain submatrices. Let A be a (0, 1)-matrix forbidding any (t+l)x(lambda+2) submatrix (lambda+2).1t0l. Assume A is m-rowed and only columns of sum t+1,t+2, horizontal ellipsis ,m-l are allowed to be repeated. Assume that A has the maximum number of columns subject to the given restrictions. Assume m is sufficiently large. Then A has each column of sum 0,1, horizontal ellipsis ,t and m-l+1,m-l+2, horizontal ellipsis ,m exactly once and, given the appropriate divisibility condition, the columns of sum t+1 correspond to a t-design with block size t+1 and parameter lambda. The proof derives a basic upper bound on the number of columns of A by a pigeonhole argument and then a careful argument, for large m, reduces the bound by a substantial amount down to the value given by design-based constructions. We extend in a few directions.
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