We study the set disjointness problem in the number-on-the-forehead model.

(i) We prove that $k$-party set disjointness has randomized and nondeterministic
communication complexity $\Omega(n/4^k)^{1/4}$ and Merlin-Arthur complexity $\Omega(n/4^k)^{1/8}.$
These bounds are close to tight. Previous lower bounds (2007-2008) for $k\geq3$ parties
were weaker than $n^{1/(k+1)}/2^{k^2}$ in all ... more >>>

We prove that the set disjointness problem has randomized communication complexity
$\Omega(\sqrt{n}/2^{k}k)$ in the number-on-the-forehead model with $k$ parties, a quadratic improvement
on the previous bound $\Omega(\sqrt{n}/2^{k})^{1/2}$. Our result remains valid for quantum
protocols, where it is essentially tight. Proving it was an open problem since 1997, ... more >>>

We consider the standard two-party communication model. The central problem studied in this article is how much one can save in information complexity by allowing an error of $\epsilon$.
For arbitrary functions, we obtain lower bounds and upper bounds indicating a gain that is of order $\Omega(h(\epsilon))$ and $O(h(\sqrt{\epsilon}))$. ... more >>>

A basic goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems $f\colon(\{0,1\}^n)^{k}\to\{0,1\}$ with $k\gg\log n$ parties. We study the problems of inner product and set disjointness and determine their randomized communication complexity for every $k\geq\log n$, showing in both cases that $\Theta(1+\lceil\log n\rceil/\log\lceil1+k/\log n\rceil)$ bits are ... more >>>