Ex1: Given C_1,C_2,..,C_n convex sets in R^d such that the intersection of every d+1 of them contain a translated copy of K where K is a convex set, show that the intersection of all of them also contains a translated copy of K.
Ex2: Show that the statement above is not true if rotations of K are allowed. You can assume that K is a line segment of length 1 in R^2.
Ex3: Given a polygon (not necessarily convex) in R^d, such that for every three points in X, there is a point in X that sees all three of them. Show that X is star-shaped, or star-convex, i.e. there is a point in X that can see all its points.
Ex4: Prove the following version of infinite Ramsey theorem: Every two edge-coloring of a complete graph on a countable number of vertices contains an infinite monochromatic clique.