  We develop and apply new combinatorial and algebraic tools to understand
multiparty communication complexity in the Number On the Forehead (NOF) model,
and related Ramsey type problems in combinatorics. We identify barriers for
progress and propose new techniques to circumvent these.
  (1) We introduce a technique for constructing independent sets in hypergraphs
via combinatorial degeneration. In particular, we make progress on the corner
problem by proving the existence of a corner-free subset of $\mathbb{F}_2^n
\times \mathbb{F}_2^n$ of size $3.16^{n-o(n)}$, which improves the previous
lower bound $2.82^n$ of Linial, Pitassi and Shraibman (ITCS 2018).
  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. As a consequence of our
construction of corner-free sets, the communication complexity of the Eval
problem over $\mathbb{F}_2^n$ is at most $0.34n + O(\log n)$, which improves
the previous upper bound $0.5n + O(\log n)$.
  (2) We point out how induced matchings in hypergraphs pose a barrier for
existing tensor tools (like slice rank, subrank, analytic rank, geometric rank
and G-stable rank) to effectively upper bound the size of independent sets in
hypergraphs. This implies a barrier for these tools to effectively lower bound
the communication complexity of the Eval problem over any group $G$.
  (3) We introduce the symmetric subrank of tensors as a proposal to circumvent
the induced matching barrier and we introduce the symmetric quantum functional
as a symmetric variation on the quantum functionals (STOC 2018). We prove that
Comon's conjecture holds asymptotically for the tensor rank, the subrank and
the restriction preorder, which implies a strong connection between Strassen's
asymptotic spectrum of tensors and the asymptotic spectrum of symmetric
tensors.
