Some results on intersecting families of subsets

Abstract For an n -tuple t = ( t 1 , t 2 ,…, t n ) of integers satisfying 1⩽t 1 ⩽t 2 ···⩽t n , T( t )=T denotes the ranked partially ordered set consisting of n- tuples a = (a 1 ,a 2 ,…,a n ) of integers satisfying t n − t i ⩽ a i ⩽ t n , i = 1,2,…, n , partially ordered by defining a to precede c if a i = c i or c i = t n for i = 1,2,…, n . The rank r( a ) of a is |{ i | a i = t n }|. For 0⩽ l ⩽ n , the set consisting of all elements of rank l is called the l th rank and is denoted T l . Let b , l and m denote positive integers satisfying b ⩽ l ⩽ n and m ⩽| T l |. For a subset A of T l , Δ b A denotes the elements of T l - b which precede at least one element of A . An algorithm is given for calculating min | Δ b A |, where the minimum is taken over all m -element subsets A of T l . If t 1 = t 2 = ··· = t n = 1, it reduces to the Kruskal-Katona algorithm.