A note on multiparty communication complexity and the Hales-Jewett theorem

As a simple application we prove a lower bound on cn,k, similar to the lower bound in [19] which is roughly cn,k/kn≥exp⁡(−O(log⁡n)1/⌈log2⁡k⌉). This lower bound follows from a protocol for Partn,k. It is interesting to better understand the communication complexity of Partn,k as this will also lead to the better understanding of the Hales-Jewett number. The main purpose of this note is to motivate this study.