A problem with sets

by dapet
Tags: sets
 P: 11 Let's imagine two sets A = {1,2,...,k} and B = {-1,-2,...,-k} for some natural k, then let's create n two-element sets X_1,X_2,...,X_n such that for each 0|b|. We know how do sets X_i look like and according to this we will choose the set C = {c_1,c_2,...,c_k} where |c_i|=i such that the number (denoted MAX) of sets X_1,X_2,...,X_n that have at least one common element with C is maximal. Determine the maximal constant 0<=c<=1 such that MAX>=[cn] for arbitrary n,k and sets X_1,X_2,...,X_n. NOTE: [x] denotes the integral part of number x Example: k = 2, A = {1,2}, B = {-1,-2} n = 4, X_1 = {-1,-2}, X_2 = {-1,2}, X_3 = {1,-2}, X_4 = {1,2} we can choose C = {1,2} (in this case we have more possibilities) the number X_i that have at least one common element with C is 3, X_1 and C have no common element. From this example we can easily see, that c<=3/4, I think that c=3/4 is sufficient condition, but I can't prove it. Could anybody help me with it? Thanks.