Multiple cross-intersecting families of signed sets

A k-signed r-set on [ n ] = { 1 , … , n } is an ordered pair ( A , f ) , where A is an r-subset of [ n ] and f is a function from A to [ k ] . Families A 1 , … , A p are said to be cross-intersecting if any set in any family A i intersects any set in any other family A j . Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [ n ] . Our aim is to generalise Hiltonʹs bound to one for families of k-signed r-sets on [ n ] . The main tool developed is an extension of Katonaʹs cyclic permutation argument.