The optimal upper bound on the MP-ratio for quaternary words

The so-called MP-ratio is a kind of measure of how “packed with palindromes” a given word is. The lower bound on the MP-ratio for the set of all n-ary words is (trivially) 1, while the best possible upper bound is an open problem in the general case. It is solved for $n=2$ (where the optimal upper bound is 4) and for $n=3$ (where the optimal upper bound is 6). Also, it is known that in the n-ary case the optimal bound is between 2n and the order of growth $n{2}^{\frac{n}{2}}$. In this article we solve this problem for quaternary words, for which we show that the best possible upper bound on the MP-ratio equals 8. We believe that this is the last case in which the result is 2n, that is, we believe that for $n⩾5$ there are words whose MP-ratio is strictly larger than 2n.