Transversal spaces

Abstract

Let A = (A₁,…, A _n ) be a family of subsets of a given set E. A subset {x_1,…, x_n}≠ of E such that x_i ∈ A_i for each i (1≤ i≤ n) is called a transversal of A. Of course, not every family possesses a transversal. For example, the family

$$(\{ a,b,c\} ,\{ d,e\} ,\{ a,c\} )$$

(where a, b, c, d, e are assumed distinct) has several transversals, one being {a, b, c, d}, with (for instance) a∈{a, b, c},b∈ {a, b}, d ∈ {d, e} and c∈{a,c}; whereas the family

$$(\{ a,b\} ,\{ a,b\} ,\{ a\} )$$

has none. A partial transversal of A of length / is a transversal of a subfamily of/ sets of A So, for example, the latter family above has a partial transversal {a, b} of length 2, this being a transversal of ({a, b}, {a}).