## Graduate Student Seminar

**Arithmetic structures in random sets** - by Mariah Hamel.

In this talk, we will discuss additive properties of dense subsets of random sets. Sarkozy's theorem for squares states that for any fixed positive number delta there is a large integer N_0 such that if N > N_0 and if A{1,...,N} with |A| >= delta N, then A must contain two elements whose difference is a perfect square. We extend this result to a random setting, where instead of requiring that A have positive relative density in {1,...,N}, we require that A has positive relative density in a random set. If time permits we will discuss a similar result which ensures long arithmetic progressions in sumsets of subsets of random sets. This is joint work with Izabella Laba.