Model Theory, Spring 2001 Problem Set 1: due Jan. 23

Marker: 5.5 problems 1-7; we will discuss which ones should be turned in. I think 2 and 3 were done last semester.

1. (Ramsey) Let G be a group which can be written as a union of n cosets X = {H_{i}a_{i} : i < n} but not the union of any proper subset of X. Then the intersection of the H_{i} has index at most n! in G and in particular each H_{i} has finite index.

2. (Erdos-Rado) Let G be a group which can be written as a union of λ cosets, X = {H_{i}a_{i} : i < λ} but not the union of any proper subset of X. Show the index of H_{0} in G is at most 2^{λ}.