A Note on Universal and Canonically Coloured Sequences

Abstract

A sequence X = {xi}ni=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-colouring χ of the sequence X is canonical if χ(xi) = χ(xj) whenever xi = xj. We prove that for any fixedt the length of the shortest sequence over an alphabet of size t, which has the property that every r-colouring of its entries contains a t-universal and canonically coloured subsequence, is at most $cr^{\lfloor\frac{t}{2}\rfloor}$. This is best possible up to a multiplicative constant c independent of r.