We study the k-party "number on the forehead" communication complexity of composed functions $${f \circ \vec{g}}$$f?g?, where $${f:\{0,1\}^n \to \{\pm 1\}}$$f:{0,1}n?{±1}, $${\vec{g} = (g_1,\ldots,g_n)}$$g?=(g1,?,gn), $${g_i : \{0,1\}^k \to \{0,1\}}$$gi:{0,1}k?{0,1} and for $${(x_1,\ldots,x_k) \in (\{0,1\}^n)^k}$$(x1,?,xk)?({0,1}n)k, $${f \circ \vec{g}(x_1,\ldots,x_k) = f(\ldots,g_i(x_{1,i},\ldots,x_{k,i}), \ldots)}$$f?g?(x1,?,xk)=f(?,gi(x1,i,?,xk,i),?). When $${\vec{g} = (g, g,\ldots, g)}$$g?=(g,g,?,g), we denote $${f \circ \vec{g}}$$f?g? by $${f \circ g}$$f?g. We show that there is an $${O({\rm log}^3 n)}$$O(log3n) cost simultaneous protocol for SYM $${\circ g}$$?g when k > 1 + log n, SYM is any symmetric function and g is any function. When k > 1 + 2 log n, our simultaneous protocol applies to SYM $${\circ \, \vec{g}}$$?g? with $${\vec{g}}$$g? being a vector of n arbitrary functions. We also get a non-simultaneous protocol for SYM $${\circ \, \vec{g}}$$?g? of cost $${O(n/2^k \cdot {\rm log}\, n+ k {\rm log}\, n)}$$O(n/2k·logn+klogn) for any k ? 2. In the setting of k ≤ 1 + log n, we study more closely functions of the form MAJORITY $${\circ g}$$?g, MOD m$${\circ g}$$?g and NOR $${\circ g}$$?g, where the latter two are generalizations of the well-known and studied functions generalized inner product and disjointness, respectively. We characterize the communication complexity of these functions with respect to the choice of g. In doing so, we answer a question posed by Babai et al. (SIAM J Comput 33:137---166, 2003) and determine the communication complexity of MAJORITY?QCSB k, where QCSB k is the "quadratic character of the sum of the bits" function. In the second part of our paper, we utilize the connection between the `number on the forehead' model and Ramsey theory to construct a large set without a k-dimensional corner (k-dimensional generalization of a k-term arithmetic progression) in $${(\mathbb{F}_{2}^{n})^k}$$(F2n)k, thereby obtaining the first non-trivial bound on the corresponding Ramsey number. Furthermore, we give an explicit coloring of [N] × [N] without a monochromatic two-dimensional corner and use this to obtain an explicit three-party protocol of cost $${O(\sqrt{n})}$$O(n) for the EXACTN function. For x1,x2,x3n-bit integers, EXACTN(x1,x2,x3) = ?1 iff x1 + x2 + x3 = N.