A problem of A. Sarkozy on coloured versions of Lagrange and Vinogradov's theorems.

In this talk we describe some progress on the following problem of A. Sarkozy. For any integer $K\geq 1$, let $t(K)$ be the smallest integer such that when the set of squares is coloured in one of $K$ colours, every sufficiently large integer can be written as a sum of at most $s(K)$ squares. Also, let $t(K)$ be the corresponding integer in the analogous context for the set of primes. The problem is to find optimal upper bounds for $s(K)$ and $t(K)$ in terms of $K$.