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.