(a) Suppose F is a family of subsets of {1, 2,..., 60}, and that |x| = 3 for all x ∈ F. Assume further that x ∩ y = ∅ for every x, y ∈ F. What is the largest possible value of |F|?
(b) For any integer k ≥ 1, a discrete interval of length k is defined as a set of k consecutive integers. For example, {4, 5, 6, 7} and {−8, −7, −6, −5} are discrete intervals of length four, and {101, 102, 103} is a discrete interval of length three. A family F of discrete intervals of length k is called intersecting if x ∩ y ≠ ∅ for all x, y ∈ F. Describe a largest possible intersecting family F of discrete intervals of length 240, say how large it is, and prove that it is largest possible.
I am not entirely sure how to approach either of these problems. For part a, am I supposed to use the inclusion and exclusion principle or a set of Venn diagrams? Am I supposed to check for different subsets and then try to find the largest about of subsets in $F$?
And for part b, am I supposed to find a set of subsets of length 240 that all have common elements? I am not entirely sure about how to approach how to do that. Once again, will it have to involve drawing a sett of Venn diagrams and further manipulation?
Any help?
For (a), there's 60 numbers available, each subset has exactly 3 of them, and no two subsets contain the same number. So there can be at most 60/3=20 subsets in the family. You could dress it up in set theory notation for a more formal proof but I don't see the benefit.
For (b), you want 240 consecutive intervals, starting from the numbers 1 to 240. They all have non-zero intersection because they all include the number "240" itself. It's maximal because if you have 241 intervals then at least one pair must differ in starting value by at least 240, and that pair will not overlap.