  There is a large and important collection of Ramsey-type combinatorial
problems, closely related to central problems in complexity theory, that can be
formulated in terms of the asymptotic growth of the size of the maximum
independent sets in powers of a fixed small (directed or undirected)
hypergraph, also called the Shannon capacity. An important instance of this is
the corner problem studied in the context of multiparty communication
complexity in the Number On the Forehead (NOF) model. Versions of this problem
and the NOF connection have seen much interest (and progress) in recent works
of Linial, Pitassi and Shraibman (ITCS 2019) and Linial and Shraibman (CCC
2021).
  We introduce and study a general algebraic method for lower bounding the
Shannon capacity of directed hypergraphs via combinatorial degenerations, a
combinatorial kind of "approximation" of subgraphs that originates from the
study of matrix multiplication in algebraic complexity theory (and which play
an important role there) but which we use in a novel way.
  Using the combinatorial degeneration method, we make progress on the corner
problem by explicitly constructing a corner-free subset in $F_2^n \times F_2^n$
of size $\Omega(3.39^n/poly(n))$, which improves the previous lower bound
$\Omega(2.82^n)$ of Linial, Pitassi and Shraibman (ITCS 2019) and which gets us
closer to the best upper bound $4^{n - o(n)}$. Our new construction of
corner-free sets implies an improved NOF protocol for the Eval problem. In the
Eval problem over a group $G$, three players need to determine whether their
inputs $x_1, x_2, x_3 \in G$ sum to zero. We find that the NOF communication
complexity of the Eval problem over $F_2^n$ is at most $0.24n + O(\log n)$,
which improves the previous upper bound $0.5n + O(\log n)$.
