An Application of Hindman's Theorem to a Problem on Communication Complexity

We consider the $k$-party communication complexity of the problem of determining if a word $w$ is of the form $w_0a_1w_1a_2\dots w_{k-1}a_kw_k$, for fixed letters $a_1,\dots,a_k$. Using the well-known theorem of Hindman (a Ramsey-type result about finite subsets of natural numbers), we prove that for $k=4$ and $5$ the communication complexity of the problem increases with the length of the word $w$.