On t-intersecting families of signed sets and permutations

Abstract

a b s t r a c tA family A of sets is said to be t-intersecting if any two sets in A contain at least t common elements. A t-intersecting family is said to be trivial if there are at least t elements common to all its sets.Let X be an r-set {x 1 , . . . , x r }. For k ≥ 2, we define S X ,k and S * X ,k to be the families of k-signed r-sets given by

The Hilton-Spencer cycle theorems via Katona's shadow intersection theorem

“…Then, C\{x} = B = f i (A i ) for some C ∈ A 0 , x ∈ C, and A i ∈ A i . Since 1 ≤ i ≤ p, (4) gives us C ∩ A i = ∅, but this contradicts (5). The claim follows.…”

“…It is worth pointing out that the strict inequality is only used for (4), from which we obtain Claim 7. Let A be an intersecting subfamily of I G (4) gives us A i ∩ A j = ∅, but this contradicts (5). Therefore,…”