Uniform Intersecting Families with Covering Number Four

Let X be a finite set. We denote by (~) the family of all k-element subsets of X. A family ~satisfying ~ c (~) is called k-uniform. The vertex set of @ is X and it is often denoted by V(Y). An element of ~ is also called an edge of J~. The family ~ is called intersecting if Fc~ G ¢ 25 holds for every F, G ~ ~ . A set C c X is called a cover (or transversal set) of Y if it intersects every edge of ~ ' . A cover C is also called a t-cover if [C[ = t . The set of all t-covers of ~ is denoted by cg,(~-). The covering n u m b e r of Y is the minim u m cardinal i ty of the covers and is denoted by z ( ~ ) . By the definition, r ( ~ ) = min{ t : c ~ , ( ~ ) # ~ } .