There is also an infinite version of Helly's theorem for compact sets.
Definition of hyperplane and halfspace.
Application of Helly's theorem: Centerpoint theorem.
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.
Ramsey for more colors and hypergraphs, e.g. R_1(a,b)=a+b-1
Thm: R_t(k1,k2,..,kc)<= 1+ R_{t-1}(R_t(k1-1,k2,..,kc),..,R_t(k1,..,kc-1))
Our bound for R_3(k,k) is tower in k...
In fact R_3(k,k) is between 2^\Theta(k^2) and 2^2^\Theta(k).
We also know R_2(3,k)=\Theta(k^2/log k).
Also big theory of infinite Ramsey - if finite number of colors, exists infinite clique in infinite graph.
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.
Hw2: Consider a complete graph on the real numbers as vertices. Show that there exists a colouring of its edges with the natural numbers (every color is an integer) such that there does not exist a monochromatic K_3 (clique of three vertices).