Larger Corner-Free Sets from Better NOF Exactly-N Protocols

A subset of the integer planar grid [N] x [N] is called corner-free if it contains no triple of the form (x,y), (x+ 3,y), (x,y+ 3). It is known that such a set has a vanishingly small density, but how large this density can be remains unknown. The only previous construction, and its variants, were based on Behrend's large subset of [N] with no 3-term arithmetic progression. Here we provide the first construction of a corner-free set that does not rely on a large set of integers with no arithmetic progressions. Our approach to the problem is based on the theory of communication complexity. In the 3-players exactly-N problem the players need to decide whether x + y + z = N for inputs x,y,z and fixed N. This is the first problem considered in the multiplayer Number On the Forehead (NOF) model. Despite the basic nature of this problem, no progress has been made on it throughout the years. Only recently have explicit protocols been found for the first time, yet no improvement in complexity has been achieved to date. The present paper offers the first improved protocol for the exactly-N problem.