  A subset of the integer planar grid $[N] \times [N]$ is called corner-free if
it contains no triple of the form $(x,y), (x+\delta,y), (x,y+\delta)$. It is
known that such a set has a vanishingly small density, but how large this
density can be remains unknown. The best previous construction was based on
Behrend's large subset of $[N]$ with no $3$-term arithmetic progression. Here
we provide the first substantial improvement to this lower bound in decades.
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. This is also
the first significant example where algorithmic ideas in communication
complexity bear fruit in additive combinatorics.
