Localisation-Resistant Random Words With Small Alphabets

We consider q-coloured words, that is words on {1, ..., q} where no two consecutive letters are equal. Motivated by multipartite colouring games with nonsignalling resources, we are interested in random q-coloured words satisfying a k-localisability property. More precisely, the probability of containing any given pair of words as subwords spaced at least k letters apart can depend only on their lengths. We focus on the issue of the smallest alphabet size q for which a probability distribution for such random words can exist. For k = 1, we prove a lower bound of q >= 4. The bound is optimal because there exists a suitable distribution for random 4-colourings that was constructed by Holroyd and Liggett in 2015. Our lower bound can be generalized to k-localisable random words where the letters of each subword of k +1 letters must be pairwise different. We show that the alphabet size in this case must be at least (k + 1) . (1 + 1/k)(k).