Arithmetic Combinatorics

Let A be subset of {1,...,n}. We say that A is square sum-free if the sum of any two different elements of A is not a square. Erdos and Sarkozy asked whether a square sum-free set can have more than n(1/3+epsilon) elements (motivated by the sequence 1,4,7, 10...). Answering this, Massias constructed a square sum-free set of density 11/32. In this talk, we show that 11/32 is, in fact, the sharp bound. Joint work with A. Khalfalah.