Intersecting systems of signed sets

A family ℱ of sets is said to be (strictly] EKR if no non-trivial intersecting sub-family of ℱ is (as large as) larger than some trivial intersecting sub-family of ℱ. For a finite set X := {x1,..., x|X|} and an integer k ≥ 2, we define SX,k to be the family of signed sets given by SX,k := {{(x 1,a1),...,(x|X|a|X|)} : a i ε [k], i = 1,..., |X|}. For a family ℱ, we define S ℱ,k := ∪ℱεℱ S ℱk. We conjecture that for any ℱ and k ≥ 2, Sℱ,k is EKR, and strictly so unless k = 2 and ℱ has a particular property. A well-known result (stated by Meyer and proved in different ways by Deza and Frankl, Engel, Erdos et al., and Bollobás and Leader) supports this conjecture for ℱ = (r[n]). The main theorem in this paper generalises this result by establishing the truth of the conjecture for families ℱ that are compressed with respect to some f* ε ∪ℱεℱ (i.e f ε ℱ ε ℱ, f* ∉ ℱ ⇒ (ℱ/{f}) ∪ {f*} ε ℱ). We also confirm the conjecture for families ℱ that are uniform and EKR.peer-reviewe