Maximizing Sets: A Problem with Sets

[dapet]

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<i=<n X_i = {a,b} where a is from A and b is from B but |a|<>|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.

 

[AKG]

This doesn't make a whole lot of sense. For one:

we will choose the set C = {c_1,c_2,...,c_k} where |c_i|=i

To me, this says that C = {1, 2, 3, 4, ..., k}. It seems that the set C will always look like this. From there, it is still not clear as to exactly what we're looking for. It is further confused by your example because you said:

for each 0<i=<n X_i = {a,b} where a is from A and b is from B

But then gave, as an example:

X_4 = {1,2}

which takes both its elements from A.