Algorithm: find largest subset from arrays

Question

I have multiple arrays. I need to find the largest subset of arrays, such that, all arrays in that subset have atleast an element in common with each other. By largest I mean, the subset should have most number of arrays. I am not interested in finding which particular arrays are in the subset, but rather the size of the subset. eg if:

a1 = [1,3,7]
a2 = [3,5,7]
a3 = [2,8,9]
a4 = [7,8,9]

then I should get largest subset size as 3, because largest subset of given arrays would be a1 , a2 and a4 , because:
a1 ∩ a2 != ∅ && a1 ∩ a4 != ∅ && a2 ∩ a4 != ∅

I have a function common(array1,array2) which returns true if array1 ∩ array2 != ∅ and false otherwise.
One way of solving it would be to make all possible pairs of arrays, and check them for commonality. But the issue here is, given a list of pairs that have common element(s) between them, how to construct the largest subset.
eg given the above example, how to construct {a1,a2,a4} from (a1,a2), (a1,a4), (a2,a4), (a3,a4).

Answer 1

Since you are not interested in finding which particular arrays are in the subset, but rather only the size of the subset, one way would be to create a map of all the possible values to the number of arrays containing that value.

For the example in the question, the map would look something like:

count[1] = 1 // contained by a1
count[2] = 1 // contained by a3
count[3] = 2 // contained by a1, a2
count[7] = 3 // contained by a1, a2, a4
count[8] = 2 // contained by a3, a4
count[9] = 2 // contained by a3, a4

The highest value in the count map (in this case, 3 ) is the desired result.