  The ExactlyN problem in the number-on-forehead (NOF) communication setting
asks $k$ players, each of whom can see every input but their own, if the $k$
input numbers add up to $N$. Introduced by Chandra, Furst and Lipton in 1983,
ExactlyN is important for its role in understanding the strength of randomness
in communication complexity with many players. It is also tightly connected to
the field of combinatorics: its $k$-party NOF communication complexity is
related to the size of the largest corner-free subset in $[N]^{k-1}$.
  In 2021, Linial and Shraibman gave more efficient protocols for ExactlyN for
3 players. As an immediate consequence, this also gave a new construction of
larger corner-free subsets in $[N]^2$. Later that year Green gave a further
refinement to their argument. These results represent the first improvements to
the highest-order term for $k=3$ since the famous work of Behrend in 1946. In
this paper we give a corresponding improvement to the highest-order term for
all $k>3$, the first since Rankin in 1961. That is, we give a more efficient
protocol for ExactlyN as well as larger corner-free sets in higher dimensions.
  Nearly all previous results in this line of research approached the problem
from the combinatorics perspective, implicitly resulting in non-constructive
protocols for ExactlyN. Approaching the problem from the communication
complexity point of view and constructing explicit protocols for ExactlyN was
key to the improvements in the $k=3$ setting. As a further contribution we
provide explicit protocols for ExactlyN for any number of players which serves
as a base for our improvement.

Computational Complexity;Combinatorics