We study the k-party `number on the forehead' communication complexity of composed functions fg, where f:01n1 , g=(g1gn), gi:01k01 and for (x1xk)(01n)k , fg(x1xk)=f(gi(x1ixki)). When g=(ggg) we denote fg by fg. We show that there is an O(log3n) cost simultaneous protocol for Symg when 1+\log n">k1+logn, Sym is any symmetric function and g is any function. When 1 + 2 \log n">k1+2logn, our simultaneous protocol applies to Symg with g being a vector of any n functions. We also get a non-simultaneous protocol for Symg of cost O(n2klogn+klogn) for any k2. In the setting of k1+logn we study more closely functions of the form Majorityg, Modmg, and Norg, 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 Journal on Computing, 33:137--166, 2003) and determine the communication complexity of MajorityQcsbk, where Qcsbk 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 (Fn2)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 2-dimensional corner and use this to obtain an explicit 3-party protocol of cost O(n) for the ExactN function. For x1x2x3 n-bit integers, ExactN(x1x2x3)=−1 iff x1+x2+x3=N.