Arindam-thm: Given a convex set K and X_1,..X_n convex sets in R^d such that for every d+1 sets there exists a translation of K that intersects with all of the d+1 sets, then there is a translation of K which intersects with all X_i's. Taking K to be a point gives the original Helly's theorem, while taking it to be a circle gives a solution to the homework problem.
Proof: Using Helly and Minkowski-sum: A+B={a+b: a\in A, b\in B}.
Convex set Erdos-Szekeres: Using Klein's from any 5 there are 4 in convex pos. (if general position)
n(k): from any n we have k in convex pos. Existence follows from Ramsey: color a four point subset blue if in non-convex position and red if in convex position. A blue K^4_5 cannot exist.
Better bound with the cups and caps method: f(k,l) smallest number of points in the plane such that either a k-cup or a l-cap exists.
n(k)<=f(k,k) and f(k,l)<= {k+l-4 \choose k-2)+1 and this is sharp, to be proved next class.
HW3: Given an n*n grid, how many points are in convex, gen position? Give any upper/lower bounds you can come up with.