Homework for combinatorics 1 (part 2)
(1) Let be a set of at least points in . Prove that can be partitioned into two subsets whose convex hulls intersect.
(2) Let be convex sets in such that any of them intersect. Prove that all of them intersect.
(3) Let be a fixed prime. Construct a family of subsets of such that each has cardinality but any two has intersection
either 0, 1, or .
(4)* Let be a family of subsets of such that the intersection of any three is even. Prove that for all sufficiently large , .
(*) is a bonus problem, it does not count if you cannot do it, but if you can, I am willing to hear the solution.