A Hilton–Milner-type theorem and an intersection conjecture for signed sets

A family A of sets is said to be intersecting if any two sets in A intersect (i.e. have at least one common element). A is said to be centred if there is an element common to all the sets in A ; otherwise, A is said to be non-centred. For any r ∈ [ n ] : = { 1 , … , n } and any integer k ≥ 2 , let S n , r , k be the family { { ( x 1 , y 1 ) , … , ( x r , y r ) } : x 1 , … , x r  are distinct elements of  [ n ] ,  y 1 , … , y r ∈ [ k ] } of k -signed r -sets on [ n ] . Let m : = max { 0 , 2 r − n } . We establish the following Hilton–Milner-type theorems, the second of which is proved using the first: A 1 and A 2 are non-empty cross-intersecting (i.e. any set in A 1 intersects any set in A 2 ) sub-families of S n , r , k , then | A 1 | + | A 2 | ≤ n r k r − ∑ i = m r r i ( k − 1 ) i n − r r − i k r − i + 1 . (ii) If A is a non-centred intersecting sub-family of S n , r , k , 2 ≤ r ≤ n , then | A | ≤ { n − 1 r − 1 k r − 1 − ∑ i = m r − 1 r i ( k − 1 ) i n − 1 − r r − 1 − i k r − 1 − i + 1 if  r < n ; k r − 1 − ( k − 1 ) r − 1 + k − 1 if  r = n . We also determine the extremal structures. (ii) is a stability theorem that extends Erdős–Ko–Rado-type results proved by various authors. We then show that (ii) leads to further evidence for an intersection conjecture suggested by the author about general signed set systems.