A combinatorial problem involving graphs and matrices (original) (raw)

1982, Discrete Mathematics

In this paper we discuss a combinatorial problem involving graphs and matrims Our problr::m is a matrix analogue of the classical problem of finding a system of distinct representatilYes (transversal) of a family of sets and relates closely to an extremal problem involving l-f actors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n 3 1, let II denote the n element set (1,2,3,.. . , n}. Then let A be a k x t matrix. We say t?tat A satisfies property .P(n, k) when the following condition is satisfied: For every k-ttiple (x+2,.. . ,x&ok., there exist k distinct integers jr, j2,.. . , jk so that xi = aiji for 'i = 1,2,..., k. The minimum value of t for which there exists a k x t matrix A satisfying property P(n, k) is denoted by f(n, k). For each k > 1 and n sufficiently large, we give an explicit formula for f(n, k); for each n * 1 find k sufficiently large, we use probabilistic methods to provilde inequalities for F(n, k). Let * = (Ai : 3 s i 2s k} be an indexed family of sets. .4 ser S = (sl: s2,. .. , sk} of k distinct elements is called :i system of distinct representatives (SDR) of 9 when SiEAifori=l,2,..., k. The following well-known theorem of F*. Hall [2] gives a necessary and sufficient condition for the existence of a SDR o!! a family 3.