A note on the universal and canonically colored sequences

A sequence X = {xi}i=1 over an alphabet containing t symbols is t-universal if every permutation of those symbols is contained as a subsequence. Kleitman and Kwiatkowski showed that the minimum length of a t-universal sequence is (1−o(1))t2. In this note we address a related Ramsey-type problem. We say that an r-coloring χ of the sequence X is canonical if χ(xi) = χ(xj) whenever xi = xj . We prove that for any fixed t the length of the shortest sequence over an alphabet of size t, which has the property that every r-coloring of its entries contains a t-universal and canonically colored subsequence, is at most crb t 2 c. This is the best possible up to a multiplicative constant c independent of r.